5.1: Sonata Form - Introduction to Sonata Theory
A classical sonata is a multi-movement work for solo instrument, chamber ensemble, or orchestra with at least one movement in sonata form . Almost always, the first movement of a sonata is in sonata form. The last movement is typically an upbeat finale , which can be in a number of different forms. Inner movements (second movement of a three-movement sonata; second and third movements of a four-movement sonata) are typically slow movements and/or dance movements (minuets or scherzos). Any non-dance movement in a sonata can take sonata form, but rarely all of them at once. Commonly, only the first movement takes sonata form, or the first and one other movement. The other movements will take other standard forms, such as minuet/trio, theme-and-variations, rondo, or sonata-rondo. In this unit, we will focus on sonata forms, particularly as they are found in first movements of instrumental sonatas.
Sonata form
Sonata form shares many structural properties with small ternary form (specifically the rounded binary variant). That is, the core music of a sonata movement can be understood as exhibiting a large, three-part pattern of ABA’.
However, like small ternary, a sonata-form movement also has a two-part structure, with the first A section forming the first “half” of the movement, and the B and A’ sections working together to form the second “half.” This is most obviously seen in early sonata movements, which repeat each of those two parts. This is called a double-reprise structure, and on this level looks exactly like a rounded-binary theme.
||: A :||: B A’ :||
which sounds like
AABA’BA’
Later sonatas shed the repeat of the second reprise, only repeating the first reprise.
||: A :|| B A’ ||
which sounds like
AABA’
Still later sonatas often shed both sets of repeats, making the binary aspect of the structure much less obvious (if it even exists) than the ternary aspect of the structure.
|| A || B A’ ||
which sounds like
ABA’
The A section (or module ) exhibits what we call exposition function, and the A’ section exhibits recapitulation function. (These are the same names as other small ternary forms, like a minuet or the theme of a theme-and-variations movement.) The B section exhibits what we call development function. (This differs from the contrasting middle of other small ternary forms.) These functions will be briefly defined below and explored in greater detail in other resources.
As in minuets, it is common and acceptable to refer to the A section as “the exposition,” the B section as “the development,” and the A’ section as “the recapitulation,” as long as we retain the conceptual difference between formal modules (the passages of music) and formal functions (what those passages do in the context of the larger musical structure).
Formal functions in a sonata movement
Exposition function involves three important features:
- establish the home-key tonality;
- move to and establish a secondary key by means of a strong cadence (PAC), called the essential expositional closure (EEC); and
- lay out the melodic themes and thematic cycle that will form the foundation for the rest of the movement.
We can think about this as a question, or at least an incomplete statement: all the main melodic material of the movement is presented, but it ends in the “wrong” key.
Development function highlights the openness of this question by cycling through the same melodies in the same order, exploring the instability of the secondary key. This exploration can be accomplished by any of the following:
- beginning in the secondary key, but modulating to another key before achieving a cadence;
- employing non-functional harmonic progressions, such as harmonic sequences;
- frequent modulation; or
- multiple tonal goals other than the home key or secondary key.
It then intensifies the need for an “answer” by progressing toward an active dominant (an HC or a major arrival on V or V 7 ) in the home key.
If exposition function presents a “question,” and development function explores and then intensifies this question, recapitulation function answers the question. Recapitulation function does this by progressing one last time through the thematic cycle, but this time bringing it to a satisfactory completion: a PAC in the home key, called the essential sonata closure (ESC).
Cadential goals
As hinted at above, sonata form is anchored around several important cadences. They serve as signposts for the formal structure, as well as goals of the music leading into them. We will note these cadences using a Roman numeral for the key (relative to the home key) followed by a colon and the type of cadence.
As in small ternary form, the exposition tends to end with a PAC in a secondary key (V:PAC in a major-key movement, III:PAC or v:PAC in a minor-key movement). Rather than being simply a norm, though, this cadence is essential to the form. This PAC is called the essential expositional closure , or EEC.
The recapitulation has a corresponding cadence, also familiar from small ternary form: a PAC in the home key, which tends to correspond thematically with the EEC. This I:PAC is called the essential sonata closure , or ESC.
The development section, like the contrasting middle in small ternary form, typically ends with a HC in the home key (I:HC) or a dominant arrival in the home key, which prepares the arrival of the recapitulation.
or
Commonly, the exposition and recapitulation each have an additional cadential goal that is not shared with other small-ternary-like forms. These goals each occur between the beginning of the module and the cadential goal (EEC or ESC), and they often—though not always—involve a pause or stoppage of melodic or harmonic motion. Thus, each of these halfway cadences is called a medial caesura (MC). Both PACs and HCs, in the home key or in a secondary key, can function as medial caesurae.
Defaults and deformations
Sonata form is not a single recipe for structuring a movement. It is a set of norms that are almost always violated, or at least altered, in some way in a composition. Thus, we can talk about individual pieces being “in dialog with” sonata norms. That is, they incorporate enough of the definitive elements to be considered “in sonata form,” but they carry enough unique elements to cause some tension with the norms. Over time, some of the unique elements used by classical composers were adopted by others and became norms for a later generation. Thus, even the “typical” sonata movement is different at different times and places in history.
In general, though, for each element of sonata form, we can identify default properties of that element. Sometimes there are multiple possibilities that can be considered normative. For example, the most common cadence used for the MC of a classical sonata movement is V:HC. This is called the first-level default . The second most common MC cadence is a I:HC—the second-level default . Where multiple possibilities are normative, but one is more common or preferred over another, we use this language.
Non-default properties of a particular sonata movement are called deformations . For example, a II:IAC MC is not at all typical of classical sonatas. Thus, we would consider it a deformation. It does not mean that a II:IAC cannot function as an MC (though I can’t think of an example). However, it means that a II:IAC is a purposeful move by the composer to contradict the norm. As such, any analysis should address this deformation and attempt to explain its musical and historical significance.
Auxiliary modules
Often the exposition–development–recapitulation structure is framed by an introduction and/or a coda . Though important musical activity can take place in these modules, they are outside the sonata proper. The essential sonata elements, as well as the ones that are similar enough from piece to piece that we can generalize about them well, take place in the exposition, development, and recapitulation. Though we will address introduction and coda materials during in-class analyses, and you should address them in your own analyses, the properties of those modules are not covered by these online resources.
Further details
The remaining sonata resources on this website are largely in reference format. Rather than walk through the details in the manner of a typical textbook, they will provide you with the defining features of the elements described in as concise a manner possible. This consision will serve you well when referencing these resources quickly during analytical activities. However, it will also force you to wrestle with them in the context of real pieces in order to understand them fully. If you simply read these resources, you will not understand sonata form. You must make use of them while listening to and analyzing music in order to assimilate the information and develop the skills necessary to apply them musically.


3: The First Law of Thermodynamics
Energy is conserved in all processes, including those associated with thermodynamic systems. The roles of heat transfer and internal energy change vary from process to process and affect how work is done by the system in that process. We will see that the first law of thermodynamics explains that a change in the internal energy of a system comes from changes in heat or work. Understanding the laws that govern thermodynamic processes and the relationship between the system and its surroundings is therefore paramount in gaining scientific knowledge of energy and energy consumption.
-
- 3.1: Prelude to The First Law of Thermodynamics
- In a car engine, heat is produced when the burned fuel is chemically transformed into mostly CO₂ and H₂O, which are gases at the combustion temperature. These gases exert a force on a piston through a displacement, doing work and converting the piston’s kinetic energy into a variety of other forms—into the car’s kinetic energy; into electrical energy to run the spark plugs, radio, and lights; and back into stored energy in the car’s battery.
-
- 3.2: Thermodynamic Systems
- A thermodynamic system includes anything whose thermodynamic properties are of interest. It is embedded in its surroundings or environment; it can exchange heat with, and do work on, its environment through a boundary, which is the imagined wall that separates the system and the environment. In reality, the immediate surroundings of the system are interacting with it directly and therefore have a much stronger influence on its behavior and properties.
-
- 3.3: Work, Heat, and Internal Energy
- Positive (negative) work is done by a thermodynamic system when it expands (contracts) under an external pressure. Heat is the energy transferred between two objects (or two parts of a system) because of a temperature difference. Internal energy of a thermodynamic system is its total mechanical energy.
-
- 3.4: First Law of Thermodynamics
- Now that we have seen how to calculate internal energy, heat, and work done for a thermodynamic system undergoing change during some process, we can see how these quantities interact to affect the amount of change that can occur. This interaction is given by the first law of thermodynamics, which argues you cannot get more energy out of a system than you put into it. We will see in this chapter how internal energy, heat, and work all play a role in the first law of thermodynamics.
-
- 3.5: Thermodynamic Processes
- The thermal behavior of a system is described in terms of thermodynamic variables. For an ideal gas, these variables are pressure, volume, temperature, and number of molecules or moles of the gas. For systems in thermodynamic equilibrium, the thermodynamic variables are related by an equation of state. A heat reservoir is so large that when it exchanges heat with other systems, its temperature does not change.
-
- 3.6: Heat Capacities of an Ideal Gas
- We learned about specific heat and molar heat capacity previously; however, we have not considered a process in which heat is added. We do that in this section. First, we examine a process where the system has a constant volume, then contrast it with a system at constant pressure and show how their specific heats are related.
-
- 3.7: Adiabatic Processes for an Ideal Gas
- When an ideal gas is compressed adiabatically, work is done on it and its temperature increases; in an adiabatic expansion, the gas does work and its temperature drops. Adiabatic compressions actually occur in the cylinders of a car, where the compressions of the gas-air mixture take place so quickly that there is no time for the mixture to exchange heat with its environment.
Thumbnail: Different thermodynamic paths taken by a system in going from state A to state B. For all transitions, the change in the internal energy of the system \(ΔE_{int}=Q−W\) is the same.


4.2: Using Tools to Support Science Learning
Course Competency 4. Examine strategies that teach early science skills.
Criteria 4.2 identify tools to gather information and compare observed objects.
What are Tools?
As you learned in the previous section, children learn about science through their observations. Teachers can guide children's observations through the use of tools. Tools can be anything that teachers use to help children learn through observation, documenting their observations and discussing them. The environment and the materials in it as well as the language teachers use can be tools in preschool science.
Tools in the Environment
Language and Vocabulary as a Tool in Preschool Science
Language is a powerful tool in preschool science education, serving several important purposes:
1. **Communication:** Language allows teachers and children to communicate about scientific concepts, observations, questions, and findings. It enables children to express their ideas, thoughts, and curiosity related to science.
2. **Vocabulary Development:** Science activities and discussions introduce new vocabulary related to scientific concepts, materials, processes, and phenomena. Through exposure to these words and terms, children expand their vocabulary and develop a deeper understanding of scientific language.
3. **Conceptual Understanding:** Language helps children develop a conceptual understanding of scientific concepts. Teachers use language to explain abstract ideas, make connections between concepts, and guide children in forming mental models of scientific phenomena.
4. **Questioning and Inquiry:** Language encourages questioning and inquiry-based learning in preschool science. Teachers use open-ended questions to prompt children's curiosity, critical thinking, and exploration of scientific topics. Children learn to ask questions, make predictions, and seek answers through observation and investigation.
5. **Observation and Description:** Language is used to describe observations, experiences, and findings during science activities. Children learn to use descriptive language to communicate what they see, hear, touch, smell, and taste, enhancing their observational skills and attention to detail.
6. **Storytelling and Narratives:** Storytelling and narratives are powerful tools for conveying scientific concepts in a relatable and engaging way. Teachers use storytelling to introduce scientific topics, explain processes, and connect scientific ideas to children's everyday experiences.
7. **Collaboration and Discussion:** Language facilitates collaboration and peer-to-peer discussion in preschool science. Children engage in conversations, share ideas, compare observations, and work together to solve problems and explore scientific concepts.
8. **Documentation and Reflection:** Language supports documentation and reflection on learning experiences in preschool science. Teachers help children document their observations, experiments, and discoveries using language through drawings, written notes, or verbal reflections. This process encourages children to think critically about their learning and communicate their understanding.
Overall, language plays a fundamental role in preschool science education by promoting communication, vocabulary development, conceptual understanding, inquiry-based learning, observation and description, collaboration, storytelling, documentation, and reflection. It serves as a bridge between children's experiences, thoughts, and discoveries in the world of science. OpenAI. (2024, May 5). The role of language in preschool science education. [Chat with ChatGPT]. Retrieved from conversation with ChatGPT.
Using Scientific Vocabulary
Teachers can help children learn new vocabulary and think like scientists by using scientific vocabulary and explaining what they mean.


Skip to main content
-
-
11.1: Fixed-Axis Rotation in Rigid Bodies
-
Introduction to rotational kinematics: angular position, velocity and acceleration equations; determining angular velocity and acceleration of a point on a body rotating about a fixed axis. Includes worked examples.
-
-
11.2: Belt- and Gear-Driven Systems
-
Rotational kinematics of belt-driven pulley systems, simple gear systems, and compound gear trains. Includes worked examples.
-
-
11.3: Absolute Motion Analysis
-
Absolute motion analysis of extended bodies undergoing general planar motion. Includes worked examples.
-
-
11.4: Relative Motion Analysis
-
Relative motion analysis of extended bodies undergoing general planar motion. Includes worked examples.
-
-
11.5: Rotating Frame Analysis
-
Discussion of reference frames that translate and/or rotate with the motion of an object; derivation of kinematics equations for rotating frame analysis (analysis of multiple moving objects that are not pinned together). Includes worked examples. **Video lecture not yet available.**
-
-
11.6: Chapter 11 Homework Problems
-


7.5: Video- Consumer Attitudes and Heinz Baked Beans
Putting Consumer Attitudes and Beliefs to the Test
Just how powerful are consumer attitudes and beliefs? Are they so powerful that they can fool consumers during a taste test?
Watch the following video to see the power of consumer attitudes in action as a journalist conducts a taste test to see whether people’s brand-loyal attitudes can overrule the reality of what they are tasting.
A link to an interactive elements can be found at the bottom of this page.
You can view the transcript for “Heinz and Packaging” (opens in new window).
Contributors and Attributions
CC licensed content, Original
- Putting Consumer Attitudes and Beliefs to the Test. Authored by : Lumen Learning. License : CC BY: Attribution
CC licensed content, Shared previously
- Heinz and Packaging. Provided by : BBC. Located at : https://youtu.be/pp-6vJdYL80 . License : CC BY-NC-ND: Attribution-NonCommercial-NoDerivatives


2.3.1: The Chemical Basis of Life
-
- Last updated
- Save as PDF
- Boundless
- Boundless
Learning Objectives
- Explain the properties of carbon that allow it to serve as a building block for biomolecules
Carbon is the fourth most abundant element in the universe and is the building block of life on earth. On earth, carbon circulates through the land, ocean, and atmosphere, creating what is known as the Carbon Cycle. This global carbon cycle can be divided further into two separate cycles: the geological carbon cycles takes place over millions of years, whereas the biological or physical carbon cycle takes place from days to thousands of years. In a nonliving environment, carbon can exist as carbon dioxide (CO 2 ), carbonate rocks, coal, petroleum, natural gas, and dead organic matter. Plants and algae convert carbon dioxide to organic matter through the process of photosynthesis, the energy of light.
Carbon is Important to Life
In its metabolism of food and respiration, an animal consumes glucose (C 6 H 12 O 6 ), which combines with oxygen (O 2 ) to produce carbon dioxide (CO 2 ), water (H 2 O), and energy, which is given off as heat. The animal has no need for the carbon dioxide and releases it into the atmosphere. A plant, on the other hand, uses the opposite reaction of an animal through photosynthesis. It intakes carbon dioxide, water, and energy from sunlight to make its own glucose and oxygen gas. The glucose is used for chemical energy, which the plant metabolizes in a similar way to an animal. The plant then emits the remaining oxygen into the environment.
Cells are made of many complex molecules called macromolecules, which include proteins, nucleic acids (RNA and DNA), carbohydrates, and lipids.  The fundamental component for all of these macromolecules is carbon. The carbon atom has unique properties that allow it to form covalent bonds to as many as four different atoms, making this versatile element ideal to serve as the basic structural component, or “backbone,” of the macromolecules.
Structure of Carbon
Individual carbon atoms have an incomplete outermost electron shell. With an atomic number of 6 (six electrons and six protons), the first two electrons fill the inner shell, leaving four in the second shell. Therefore, carbon atoms can form four covalent bonds with other atoms to satisfy the octet rule. The methane molecule provides an example: it has the chemical formula CH 4 . Each of its four hydrogen atoms forms a single covalent bond with the carbon atom by sharing a pair of electrons. This results in a filled outermost shell.
Key Points
- All living things contain carbon in some form.
- Carbon is the primary component of macromolecules, including proteins, lipids, nucleic acids, and carbohydrates.
- Carbon’s molecular structure allows it to bond in many different ways and with many different elements.
- The carbon cycle shows how carbon moves through the living and non-living parts of the environment.
Key Terms
- octet rule : A rule stating that atoms lose, gain, or share electrons in order to have a full valence shell of 8 electrons (has some exceptions).
- carbon cycle : the physical cycle of carbon through the earth’s biosphere, geosphere, hydrosphere, and atmosphere; includes such processes as photosynthesis, decomposition, respiration and carbonification
- macromolecule : a very large molecule, especially used in reference to large biological polymers (e.g., nucleic acids and proteins)


26.5: Stupid Benito Cereno
Most of the texts we have read in this class have been considered National(izing) Literature, but now “Benito Cereno” can be considered a literature text beyond the nation. But to what extent does “Benito Cereno” represent “Literature Beyond the Nation”?
I feel as if I cannot confidently answer this question until I have read the whole story. Although, I can so far say I do believe this text is testing the waters with the whole “literature beyond the nation” thing.
Although Melville is American, and so are some of the characters, this book so far takes place in “Chili,” which is technically “beyond the nation.”
“In the year 1799, Captain Amasa Delano, of Duxbury, in Massachusetts,
commanding a large sealer and general trader, lay at anchor with a
valuable cargo, in the harbor of St. Maria–a small, desert, uninhabited
island toward the southern extremity of the long coast of Chili. There
he had touched for water.”
Captain Delano is traveling the coast of Chili when he runs into a stranded ship full of Spaniards and African Slaves.
This book goes beyond the nation in the aspect that it contrasts America and Europe. it doesn’t just focus on America.
Delano just keeps comparing Europe to America and how we’re so much better than these Europeans. At least that is what I took from it. Instead of focusing on America, American culture and all that it is, this story focuses a lot more on the Spaniards and their every day struggle because, “they aren’t as organized and great as Americans.” Which is pretty much what Delano is getting at.
I cant really put my points/thoughts into proper sentences so I’m going to bullet point my ideas as to why this is “beyond the nation.”
- This story doesn’t focus on a significant historical era or event in American history whereas some of other texts were based in times of war, imperialism and more.
- Every other text in this class so far, has taken place on American land at some point throughout the story, but this text (so far) isn’t on American soil at all.
- Takes place in Chili
- Focuses more on the Spaniards and African slaves rather than the American characters.
- There is no type of American imperialism and colonialism.


Skip to main content
5.1
Physical Health and Growth in Early Childhood
-
After rapid growth during infancy, physical growth in early childhood slows and becomes relatively consistent. Average height and weight differences between girls and boys are only minor in early childhood. The brain continues to grow at a rapid pace, due to overproduction of synapses as well as stimulation and increased myelination.
-
A healthy balanced diet is essential to optimal growth. Poor nutrition can have lifelong consequences.
-
Proper sleep is necessary to maintain health. Less-than-optimal sleep during early childhood is associated with overweight and obesity and may have indirect effects on other health issues in later life.
-
Vaccinations are an essential instrument against childhood illness and mortality. Accidents pose the biggest health risk to preschool children. Threats to their safety vary by the area of the world in which they live.
5.2
Motor Development and Physical Skills in Early Childhood
-
Understanding milestones in motor development helps us understand typical growth and design appropriate interventions when problems arise.
-
Children who engage in regular physical activity show enhanced cognitive development, including improved attention, self-regulation, and academic performance.
-
Children who are three and four years old should spend a minimum of three hours per day in a variety of physical activities, including at least sixty minutes of moderate intensity. Five- and six-year-olds should participate in a minimum of sixty minutes of moderate- to high-intensity activity every day.
-
Children who have physical disabilities often face obstacles and misconceptions in pursuing activities that facilitate movement.
-
Few overall differences exist between girls’ and boys’ abilities in early childhood, and they are insignificant compared to those among all boys and among all girls.
5.3
Cognition in Early Childhood
-
The stage theory of cognitive development and the sociocultural theory of cognitive development share the view that children actively construct their understanding of the world through their experiences.
-
Stage theory asserts that children acquire knowledge through their independent exploration of the environment, while sociocultural theory contends that social interactions and cultural influences play a more significant role in development.
-
The information processing model suggests that cognitive development is an ongoing process of quantitative changes that rely on a feedback loop.
-
Research into learning by observation and “pitching in” add important and complementary elements to the study of cognition in early childhood by demonstrating the ways children learn through environmental experiences with others.
5.4
Language in Early Childhood
-
Every language has unique characteristics that children must learn. These include phonemes, morphemes, semantics, syntax, grammar, and pragmatics.
-
Studying language provides a way for us to investigate cognitive processes as well as a way to understand activities related to communication and social interaction.
-
Deaf children can acquire sign language, and young children can learn multiple languages in the same manner and with the same skill as their primary languages.
-
Early literacy predicts later academic success.
5.5
Play in Early Childhood
-
Play is essential for children’s cognitive, physical, and psychosocial development and complements peer learning.
-
Mildred Parten observed that play progresses through six stages of non-social and social play.
-
Types of play vary by culture, community, and location, and all children should be afforded equal opportunities to play.
-
Socioeconomic status affects the availability of play resources and the way children play but does not necessarily result in negative physical, cognitive, or psychosocial outcomes.
-
Children with disabilities often face mobility barriers that limit opportunities for play. Early intervention programs and enhanced teacher training can help facilitate their development.


15.2: The First Law of Thermodynamics and Some Simple Processes
Learning Objectives
By the end of this section, you will be able to:
- Define the first law of thermodynamics.
- Describe how conservation of energy relates to the first law of thermodynamics.
- Identify instances of the first law of thermodynamics working in everyday situations, including biological metabolism.
- Calculate changes in the internal energy of a system, after accounting for heat transfer and work done.
One of the most important things we can do with heat transfer is to use it to do work for us. Such a device is called a heat engine . Car engines and steam turbines that generate electricity are examples of heat engines. Figure \(\PageIndex{2}\) shows schematically how the first law of thermodynamics applies to the typical heat engine.
Figure \(\PageIndex{3}\): (a) Heat transfer to the gas in a cylinder increases the internal energy of the gas, creating higher pressure and temperature. (b) The force exerted on the movable cylinder does work as the gas expands. Gas pressure and temperature decrease when it expands, indicating that the gas’s internal energy has been decreased by doing work. (c) Heat transfer to the environment further reduces pressure in the gas so that the piston can be more easily returned to its starting position.
The illustrations above show one of the ways in which heat transfer does work. Fuel combustion produces heat transfer to a gas in a cylinder, increasing the pressure of the gas and thereby the force it exerts on a movable piston. The gas does work on the outside world, as this force moves the piston through some distance. Heat transfer to the gas cylinder results in work being done. To repeat this process, the piston needs to be returned to its starting point. Heat transfer now occurs from the gas to the surroundings so that its pressure decreases, and a force is exerted by the surroundings to push the piston back through some distance. Variations of this process are employed daily in hundreds of millions of heat engines. We will examine heat engines in detail in the next section. In this section, we consider some of the simpler underlying processes on which heat engines are based.
PV Diagrams and their Relationship to Work Done on or by a Gas
A process by which a gas does work on a piston at constant pressure is called an isobaric process . Since the pressure is constant, the force exerted is constant and the work done is given as
\[W = Fd\] See the symbols as shown in Figure \(\PageIndex{4}\). Now \(F = PA\), and so \[W = PAd.\] Because the volume of a cylinder is its cross-sectional area \(A\) times its length \(d\), we see that \(Ad = \Delta V\), the change in volume; thus, \[W = P\Delta V \, (isobaric \, process).\] Note that if \(\Delta V\) is positive, then \(W\) is positive, meaning that work is done by the gas on the outside world.
(Note that the pressure involved in this work that we’ve called \(P\) is the pressure of the gas inside the tank. If we call the pressure outside the tank \(P_{ext}\), an expanding gas would be working against the external pressure; the work done would therefore be \(W = -P_{ext}\delta V\) (isobaric process). Many texts use this definition of work, and not the definition based on internal pressure, as the basis of the First Law of Thermodynamics. This definition reverses the sign conventions for work, and results in a statement of the first law that becomes \(\Delta U = Q + W\).)
It is not surprising that \(W = P\Delta V\), since we have already noted in our treatment of fluids that pressure is a type of potential energy per unit volume and that pressure in fact has units of energy divided by volume. We also noted in our discussion of the ideal gas law that \(PV\)
has units of energy. In this case, some of the energy associated with pressure becomes work.
Figure \(\PageIndex{5}\) shows a graph of pressure versus volume (that is, a \(PV\) diagram for an isobaric process. You can see in the figure that the work done is the area under the graph. This property of \(PV\) diagrams is very useful and broadly applicable: the work done on or by a system in going from one state to another equals the area under the curve on a \(PV\) diagram .
We can see where this leads by considering Figure \(\PageIndex{6}\)(a), which shows a more general process in which both pressure and volume change. The area under the curve is closely approximated by dividing it into strips, each having an average constant pressure \(P_{i(ave)}\). The work done is \(W_i = P_{i(ave)}\Delta V_i\) for each strip, and the total work done is the sum of the \(W_i\). Thus the total work done is the total area under the curve. If the path is reversed, as in Figure \(\PageIndex{6}\)(b), then work is done on the system. The area under the curve in that case is negative, because \(\Delta V\) is negative.
\(PV\) diagrams clearly illustrate that the work done depends on the path taken and not just the endpoints . This path dependence is seen in Figure \(\PageIndex{7}\)(a), where more work is done in going from A to C by the path via point B than by the path via point D. The vertical paths, where volume is constant, are called isochoric processes. Since volume is constant, \(\Delta V = 0\), and no work is done in an isochoric process. Now, if the system follows the cyclical path ABCDA, as in Figure \(\PageIndex{7}\)(b), then the total work done is the area inside the loop. The negative area below path CD subtracts, leaving only the area inside the rectangle. In fact, the work done in any cyclical process (one that returns to its starting point) is the area inside the loop it forms on a \(PV\) diagram, as Figure \(\PageIndex{7}\)(c) illustrates for a general cyclical process. Note that the loop must be traversed in the clockwise direction for work to be positive—that is, for there to be a net work output.
Example \(\PageIndex{1}\): Total Work Done in a Cyclical Process Equals the Area Inside the Closed Loop on a PV Diagram
Calculate the total work done in the cyclical process ABCDA shown in Figure \(\PageIndex{7}\)(b) by the following two methods to verify that work equals the area inside the closed loop on the \(PV\)
diagram. (Take the data in the figure to be precise to three significant figures.) (a) Calculate the work done along each segment of the path and add these values to get the total work. (b) Calculate the area inside the rectangle ABCDA.
Strategy
To find the work along any path on a \(PV\) diagram, you use the fact that work is pressure times change in volume, or \(W = P\Delta V\).
So in part (a), this value is calculated for each leg of the path around the closed loop.
Solution for (a)
The work along path AB is \[W_{AB} = P_{AB}\Delta Y_{AB}\] \[= (1.50 \times 10^6 \, N/m^2)(5.00 \times 10^{-4} \, m^3) = 750 \, J.\] Since the path BC is isochoric, \(\Delta V_{BC} = 0\), and so \(W_{BC} = 0\). The work along path CD is negative, since \(\Delta V_{CD} \) is negative (the volume decreases). The work is \[W_{CD} = P_{CD}\Delta V_{CD}\] \[= (2.00 \times 10^5 \, N/m^2)(-5.00 \times 10^{-4} \, m^3) = -100 \, J.\]
Again, since the path DA is isochoric, \(\delta V_{DA} = 0\), and so \(W_{DA} = 0\). Now the total work is \[W = W_{AB} + W_{BC} + W_{CD} + W_{DA}\] \[ = 750 \, J + 0 + (-100 \, J) + 0 = 650 \, J.\]
Solution for (b)
The area inside the rectangle is its height times its width, or \[area = (P_{AB} - P_{CD})\Delta V\] \[= [(1.50 \times 10^6 \, N/m^2) - (2.00 \times 10^5 \, N/m^2)] (5.00 \times 1-^{-4} m^3)\] \[= 650 \, J.\]
Thus, \[area = 650 \, J = W.\]
Discussion
The result, as anticipated, is that the area inside the closed loop equals the work done. The area is often easier to calculate than is the work done along each path. It is also convenient to visualize the area inside different curves on \(PV\) diagrams in order to see which processes might produce the most work. Recall that work can be done to the system, or by the system, depending on the sign of \(W\). A positive \(W\) is work that is done by the system on the outside environment; a negative \(W\) is work that is done by the system on the outside environment; a negative \(W\) represents work done by the environment on the system.
Figure \(\PageIndex{8}\)(a) shows two other important processes on a \(PV\) diagram. For comparison, both are shown starting from the same point A. The upper curve ending at point B is an isothermal process—that is, one in which temperature is kept constant. If the gas behaves like an ideal gas, as is often the case, and if no phase change occurs, then \(PV = nRT\). Since \(T\) is constant, \(PV\) is a constant for an isothermal process. We ordinarily expect the temperature of a gas to decrease as it expands, and so we correctly suspect that heat transfer must occur from the surroundings to the gas to keep the temperature constant during an isothermal expansion. To show this more rigorously for the special case of a monatomic ideal gas, we note that the average kinetic energy of an atom in such a gas is given by \[\dfrac{1}{2}m\overline v^2 = \dfrac{3}{2}kT.\] The kinetic energy of the atoms in a monatomic ideal gas is its only form of internal energy, and so its total internal energy \(U\) is \[U = N\dfrac{1}{2}m\overline v^2 = \dfrac{3}{2}NkT, \, (monatomic \, ideal \, gas),\] where \(N\) is the number of atoms in the gas. This relationship means that the internal energy of an ideal monatomic gas is constant during an isothermal process—that is, \(\Delta U = 0\). If the internal energy does not change, then the net heat transfer into the gas must equal the net work done by the gas. That is, because \(\Delta U = Q - W = 0\) here, \(Q = W\). We must have just enough heat transfer to replace the work done. An isothermal process is inherently slow, because heat transfer occurs continuously to keep the gas temperature constant at all times and must be allowed to spread through the gas so that there are no hot or cold regions.
Also shown in Figure \(\PageIndex{8}\)(a) is a curve AC for an adiabatic process, defined to be one in which there is no heat transfer—that is, \(Q = 0\). Processes that are nearly adiabatic can be achieved either by using very effective insulation or by performing the process so fast that there is little time for heat transfer. Temperature must decrease during an adiabatic expansion process, since work is done at the expense of internal energy: \[U = \dfrac{3}{2}NkT.\] (You might have noted that a gas released into atmospheric pressure from a pressurized cylinder is substantially colder than the gas in the cylinder.) In fact, because \(Q = 0, \, \Delta U = -W\) for an adiabatic process. Lower temperature results in lower pressure along the way, so that curve AC is lower than curve AB, and less work is done. If the path ABCA could be followed by cooling the gas from B to C at constant volume (isochorically), Figure \(\PageIndex{8}\)(b), there would be a net work output.
Reversible Processes
Both isothermal and adiabatic processes such as shown in Figure \(\PageIndex{8}\) are reversible in principle. A reversible process is one in which both the system and its environment can return to exactly the states they were in by following the reverse path. The reverse isothermal and adiabatic paths are BA and CA, respectively. Real macroscopic processes are never exactly reversible. In the previous examples, our system is a gas (like that in Figure \(\PageIndex{4}\)), and its environment is the piston, cylinder, and the rest of the universe. If there are any energy-dissipating mechanisms, such as friction or turbulence, then heat transfer to the environment occurs for either direction of the piston. So, for example, if the path BA is followed and there is friction, then the gas will be returned to its original state but the environment will not—it will have been heated in both directions. Reversibility requires the direction of heat transfer to reverse for the reverse path. Since dissipative mechanisms cannot be completely eliminated, real processes cannot be reversible.
There must be reasons that real macroscopic processes cannot be reversible. We can imagine them going in reverse. For example, heat transfer occurs spontaneously from hot to cold and never spontaneously the reverse. Yet it would not violate the first law of thermodynamics for this to happen. In fact, all spontaneous processes, such as bubbles bursting, never go in reverse. There is a second thermodynamic law that forbids them from going in reverse. When we study this law, we will learn something about nature and also find that such a law limits the efficiency of heat engines. We will find that heat engines with the greatest possible theoretical efficiency would have to use reversible processes, and even they cannot convert all heat transfer into doing work. The table summarizes the simpler thermodynamic processes and their definitions.
| Isobaric | Constant pressure | \(W = P\Delta V\) |
|---|---|---|
| Isochoric | Constant volume | \(W = 0\) |
| Isothermal | Constant temperature | \(Q = W\) |
| Adiabatic | No heat transfer | \(Q = 0\) |
PHET EXPLORATIONS: STATES OF MATTER
Watch different types of molecules form a solid, liquid, or gas in teh States of Matter simuator. Add or remove heat and watch the phase change. Change the temperature or volume of a container and see a pressure-temperature diagram respond in real time. Relate the interaction potential to the forces between molecules.
Summary
- One of the important implications of the first law of thermodynamics is that machines can be harnessed to do work that humans previously did by hand or by external energy supplies such as running water or the heat of the Sun. A machine that uses heat transfer to do work is known as a heat engine.
- There are several simple processes, used by heat engines, that flow from the first law of thermodynamics. Among them are the isobaric, isochoric, isothermal and adiabatic processes.
- These processes differ from one another based on how they affect pressure, volume, temperature, and heat transfer.
- If the work done is performed on the outside environment, work \((W)\) will be a positive value. If the work done is done to the heat engine system, work \((W)\) will be a negative value.
- Some thermodynamic processes, including isothermal and adiabatic processes, are reversible in theory; that is, both the thermodynamic system and the environment can be returned to their initial states. However, because of loss of energy owing to the second law of thermodynamics, complete reversibility does not work in practice.
Glossary
- heat engine
- a machine that uses heat transfer to do work
- isobaric process
- constant-pressure process in which a gas does work
- isochoric process
- a constant-volume process
- isothermal process
- a constant-temperature process
- adiabatic process
- a process in which no heat transfer takes place
- reversible process
- a process in which both the heat engine system and the external environment theoretically can be returned to their original states


16.3: Institutional Arrangements for Bases, Campuses, Parks and Ports
The institutional managers of bases, campuses, parks, and ports vary considerably among different entities and nations. Some typical arrangements are:
- Military bases are usually controlled by particular branches of the military service, such as Army, Air Force, Coast Guard, Marines or Navy.
- Campuses may be controlled by government agencies (as with the Department of Energy Laboratories), corporations, or non-profit entities (such as Dartmouth College shown in Figure 16.1.2).
- Parks may be controlled by national, state or local government agencies. Privately owned parks also exist, such as the Disney Corporation resorts.
- Ports may be controlled by corporations, government agencies or other non-profit entities.
The concepts associated with infrastructure management outlined in the first ten chapters of this book, and illustrated in Figure 16.3.1 below, apply both to independently controlled and integrated entities. Campus infrastructure managers regularly perform asset management, inventory, benchmarking, etc.
However, some regulatory differences exist for integrated entities. For example, campuses, parks and ports typically have master planning processes in place for long-term infrastructure changes. These plans may include elements such as locations and rough shapes for future buildings and designations of permanent open space. New infrastructure is explicitly tied to the entities overall mission (e.g., education and research for universities), and all new buildings or other investments would then be designed to conform to the master plan. Cities and states have similar master planning processes, but these typically recognize the dispersed decision making associated with dispersed ownership of infrastructure.
Transitions in institutional arrangements for these integrated entities can also occur, although usually over a lengthy period of time. For example, Figure 16.1.1 showed Fort Jefferson which was transferred from military control to the National Park Service. Private community and resort developers may control the entire property initially, but then control is transferred to individual property buyers and eventually a community governing organization.


1.53: Group isomorphism
A group isomorphism is a special type of group homomorphism . It is a mapping between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the respective group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic . Isomorphic groups have the same properties and the same structure of their multiplication table.
Let ( G , *) and ( H , #) be two groups, where "*" and "#" are the binary operations in G and H , respectively. A group isomorphism from ( G , *) to ( H , #) is a bijection from G to H , i . e . a bijective mapping f : G → H such that for all u and v in G one has
f ( u * v ) = f ( u ) # f ( v ).
Two groups ( G , *) and ( H , #) are isomorphic if an isomorphism between
them exists. This is written:( G , *) \(\cong\) ( H , #)
If H = G and the binary operations # and * coincide, the bijection is an automorphism .


1.7: Inverse Functions
- Refresher Topics
-
-
Simplifying Expressions
- Simplifying Expressions Involving Positive Rational Exponents
-
Solving Equations
- Solving Radical Equations (single, higher-indexed radical)
-
Simplifying Expressions
In this section, you will:
- Verify inverse functions.
- Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
- Find or evaluate the inverse of a function.
- Use the graph of a one-to-one function to graph its inverse function on the same axes.
A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.
If some physical machines can run in two directions, we might ask whether some of the function "machines" we have been studying can also run backwards. Figure \( \PageIndex{ 1 } \) provides a visual representation of this question. In this section, we will consider the reverse nature of functions.
Figure \( \PageIndex{ 1 } \): Can a function "machine" operate in reverse?
Verifying That Two Functions Are Inverse Functions
Suppose a fashion designer traveling to Milan for a fashion show wants to know what the temperature will be. He is not familiar with the Celsius scale. To get an idea of how temperature measurements are related, he asks his assistant, Betty, to convert 75 degrees Fahrenheit to degrees Celsius. She finds the formula\[ C = \dfrac{5}{9} (F−32) \nonumber \]and substitutes 75 for \(F\) to calculate\[ \dfrac{5}{9} (75−32) \approx 24^{\circ} \, \text{C}. \nonumber \]Knowing that a comfortable 75 degrees Fahrenheit is about 24 degrees Celsius, he sends his assistant the week’s weather forecast from Figure \( \PageIndex{ 2 } \) for Milan, and asks her to convert all of the temperatures to degrees Fahrenheit.
Figure \( \PageIndex{ 2 } \)
At first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her Algebra, and can easily solve the equation for \(F\) after substituting a value for \(C.\) For example, to convert 26 degrees Celsius, she could write\[ \begin{array}{rrcl}
& 26 & = & \dfrac{5}{9} (F−32) \\[6pt]
\implies & 26 \cdot \dfrac{9}{5} & = & F−32 \\[6pt]
\implies & F & = & 26 \cdot \dfrac{9}{5} +32 \approx 79 \\[6pt]
\end{array} \nonumber \]After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. She realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature.
The formula for which Betty is searching corresponds to the idea of an inverse function , which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.
Given a function \( f(x) \), we represent its inverse as \( f^{-1}(x) \), read as "\(f\) inverse of \(x\)." The raised \(−1\) is part of the notation. It is not an exponent ; it does not imply a power of \(−1\). In other words, \( f^{-1}(x) \) does not mean \( \frac{1}{f(x)} \) because \( \frac{1}{f(x)} \) is the reciprocal of \(f\) and not the inverse.
The "exponent-like" notation comes from an analogy between function composition and multiplication: just as \(a^{-1} a=1\) (1 is the identity element for multiplication) for any nonzero number \(a\), so \(f^{-1} \circ f\) equals the identity function, that is,\[( f^{-1} \circ f )(x) = f^{-1} \left( f(x) \right) = x. \nonumber \]This holds for all \(x\) in the domain of \(f\). Informally, this means that inverse functions "undo" each other. However, just as zero does not have a reciprocal, some functions do not have inverses.
Given a function \( f(x) \), we can verify whether some other function \(g(x)\) is the inverse of \(f(x)\) by checking whether either \(g(f(x))=x\) or \(f(g(x))=x\) is true. We can test whichever equation is more convenient to work with because they are logically equivalent (that is, if one is true, then so is the other.)
For example, \(y=4x\) and \(y= \frac{1}{4} x\) are inverse functions.\[( f^{-1} \circ f )(x)= f^{-1} ( 4x )= \dfrac{1}{4} ( 4x )=x \nonumber \]and\[( f \circ f^{-1} )(x)=f\left(\dfrac{1}{4}x \right)=4\left( \dfrac{1}{4}\right)=x. \nonumber \]A few coordinate pairs from the graph of the function \(y=4x\) are \( (−2, −8) \), \( (0, 0) \), and \( (2, 8) \). A few coordinate pairs from the graph of the function \(y= \frac{1}{4} x\) are \( (−8, −2) \), \( (0, 0) \), and \( (8, 2) \). If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.
For any one-to-one function \(f(x)=y\), a function \(f^{-1}( x )\) is an inverse function of \(f\) if \(f^{−1}(y)=x\). This can also be written as \(f^{-1}(f(x))=x\) for all \(x\) in the domain of \(f\). It also follows that \(f( f^{-1}(x))=x\) for all \(x\) in the domain of \(f^{-1}\) if \(f^{-1}\) is the inverse of \(f\).
The notation \(f^{-1}\) is read "\(f\) inverse." Like any other function, we can use any variable name as the input for \(f^{-1}\), so we will often write \( f^{-1}(x) \), which we read as "\(f\) inverse of \(x\)." Keep in mind that\[f^{−1}(x) \neq \dfrac{1}{f(x)} \nonumber \]and not all functions have inverses.
If for a particular one-to-one function \(f(2)=4\) and \(f(5)=12\), what are the corresponding input and output values for the inverse function?
- Solution
-
The inverse function reverses the input and output quantities, so if \( f(2)=4\), then \(f^{-1} (4)=2\); If \(f( 5 )=12\), then \(f^(-1) ( 12 )=5\). Alternatively, if we want to name the inverse function \(g\), then \(g(4)=2\) and \(g(12)=5\).
If we show the coordinate pairs from Example \( \PageIndex{ 1 } \) in a table form, the input and output are clearly reversed. See Table \( \PageIndex{ 1 } \).
| \(( x,f(x) )\) | \(( x,g(x) )\) |
|---|---|
| \(( 2,4 )\) | \(( 4,2 )\) |
| \(( 5,12 )\) | \(( 12,5 )\) |
Given that \(h^{-1} (6)=2\), what are the corresponding input and output values of the original function \( h \)?
If \(f( x )= \frac{1}{x+2}\) and \(g( x )= \frac{1}{x} −2\), is \(g= f^{-1} \)?
- Solution
-
To determine if \( g \) is the inverse function of \( f \), we just need to check if \( g \) "undoes" \( f \).\[ \begin{array}{rclcl}
g(f(x)) & = & g\left( \dfrac{1}{x + 2} \right) & \quad & \left( \text{substituting} \right) \\[6pt]
& = & \dfrac{1}{\frac{1}{x + 2}} - 2 & \quad & \left( \text{substituting} \right) \\[6pt]
& = & x + 2 - 2 & \quad & \left( \text{recognizing that }\dfrac{1}{1/a} = a \right) \\[6pt]
& = & x & \quad & \left( \text{simplifying} \right) \\[6pt]
\end{array} \nonumber \]so\[g= f^{-1} \text{ and }f= g^{-1} \nonumber \]This is enough to answer yes to the question, but we can also verify the other formula.\[ \begin{array}{rclcl}
f(g(x)) & = & f\left( \dfrac{1}{x} - 2 \right) & \quad & \left( \text{substituting} \right) \\[6pt]
& = & \dfrac{1}{\left(\frac{1}{x} + 2\right) - 2} & \quad & \left( \text{substituting} \right) \\[6pt]
& = & \dfrac{1}{\frac{1}{x}} & \quad & \left( \text{simplifying} \right) \\[6pt]
& = & x & \quad & \left( \text{recognizing that }\dfrac{1}{1/a} = a \right) \\[6pt]
\end{array} \nonumber \]
If \(f( x )= x^3 −4\) and \(g( x )= \sqrt[3]{x+4} \), is \(g= f^{-1} \)?
If \(f(x)= x^3\) (the cube function) and \(g(x)= \frac{1}{3} x\), is \(g= f^{-1} \)?
- Solution
-
\[f( g( x ) )= f\left( \dfrac{x}{3} \right) = \left( \dfrac{x}{3} \right)^3 = \dfrac{x^3}{27} \neq x. \nonumber \]No, the functions are not inverses. The correct inverse to the cube is, of course, the cube root \(\sqrt[3]{x} = x^{1/3} \), that is, the one-third is an exponent, not a multiplier.
If \(f( x )= ( x−1 )^3\) and \(g( x )= \sqrt[3]{x} +1\), is \(g= f^{-1} \)?
Finding Domain and Range of Inverse Functions
The outputs of the function \(f\) are the inputs to \(f^{-1}\), so the range of \(f\) is also the domain of \(f^{-1} \). Likewise, because the inputs to \(f\) are the outputs of \(f^{-1}\), the domain of \(f\) is the range of \(f^{-1} \). We can visualize the situation as in Figure \( \PageIndex{ 3 } \).
Figure \( \PageIndex{ 3 } \): Domain and range of a function and its inverse
When a function has no inverse function, it is possible to create a new function where that new function on a limited domain does have an inverse function. For example, the inverse of \(f(x)=\sqrt{ x }\) is \(f^{-1} (x)= x^2 \), because a square "undoes" a square root; but the square is only the inverse of the square root on the domain \([ 0, \infty )\), since that is the range of \(f(x)=\sqrt{ x } \).
We can look at this problem from the other side, starting with the square (toolkit quadratic) function \(f(x)= x^2 \). If we want to construct an inverse to this function, we run into a problem because for every given output of the quadratic function, there are two corresponding inputs (except when the input is 0). For example, the output \( 9 \) from the quadratic function corresponds to the inputs \( 3 \) and \( –3 \). But an output from a function is an input to its inverse; if this inverse input corresponds to more than one inverse output (input of the original function), then the "inverse" is not a function at all!
To put it differently, the quadratic function is not a one-to-one function; it fails the Horizontal Line Test, so it does not have an inverse function. In order for a function to have an inverse, it must be a one-to-one function.
In many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-to-one. For example, we can make a restricted version of the square function \(f(x)= x^2\) with its domain limited to \([ 0, \infty )\), which is a one-to-one function (it passes the Horizontal Line Test) and which has an inverse (the square-root function).
If \(f(x)= ( x−1 )^2\) on \([ 1, \infty )\), then the inverse function is \(f^{-1} (x)=\sqrt{ x } +1\).
- The domain of \(f\) = range of \(f^{-1}\) = \([ 1, \infty )\).
- The domain of \(f^{-1}\) = range of \(f\) = \([ 0, \infty )\).
Is it possible for a function to have more than one inverse?
No. If two supposedly different functions, say, \(g\) and \(h\), both meet the definition of being inverses of another function \(f\), then you can prove that \(g=h\). We have just seen that some functions only have inverses if we restrict the domain of the original function. In these cases, there may be more than one way to restrict the domain, leading to different inverses. However, on any one domain, the original function still has only one unique inverse.
The range of a function \(f(x)\) is the domain of the inverse function \(f^{-1} (x)\), and the domain of \(f(x)\) is the range of \(f^{-1} (x)\).
Identify which of the toolkit functions besides the quadratic function are not one-to-one, and find a restricted domain on which each function is one-to-one, if any. The toolkit functions are reviewed in Table \( \PageIndex{ 2 } \). We restrict the domain in such a fashion that the function assumes all \( y \)-values exactly once.
| Constant | Identity | Quadratic | Cubic | Reciprocal |
| \(f(x)=c\) | \(f(x)=x\) | \(f(x)= x^2\) | \(f(x)= x^3\) | \(f(x)= \frac{1}{x}\) |
| Reciprocal squared | Cube root | Square root | Absolute value | |
| \(f(x)= \frac{1}{x^2}\) | \(f(x)= \sqrt[3]{x}\) | \(f(x)=\sqrt{ x }\) | \(f(x)=| x |\) |
- Solution
-
The constant function is not one-to-one, and there is no domain (except a single point) on which it could be one-to-one, so the constant function has no meaningful inverse.
The absolute value function can be restricted to the domain \([ 0, \infty )\), where it is equal to the identity function.
The reciprocal-squared function can be restricted to the domain \(( 0, \infty )\).
We can see that the absolute value and reciprocal-squared functions (if unrestricted) are not one-to-one by looking at their graphs, shown in Figure \( \PageIndex{ 4 } \). They both would fail the Horizontal Line Test. However, if a function is restricted to a certain domain so that it passes the Horizontal Line Test, then in that restricted domain, it can have an inverse.
Figure \( \PageIndex{ 4(a) } \): Absolute value
Figure \( \PageIndex{ 4(b) } \): Reciprocal squared
The domain of function \(f\) is \((1, \infty )\) and the range of function \(f\) is \((− \infty ,−2)\). Find the domain and range of the inverse function.
Finding and Evaluating Inverse Functions
Once we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a complete representation of the inverse function in many cases.
Inverting Tabular Functions
Suppose we want to find the inverse of a function represented in table form. Remember that the domain of a function is the range of the inverse and the range of the function is the domain of the inverse. So we need to interchange the domain and range.
Each row (or column) of inputs becomes the row (or column) of outputs for the inverse function. Similarly, each row (or column) of outputs becomes the row (or column) of inputs for the inverse function.
A function \(f(t)\) is given in Table \( \PageIndex{ 3 } \), showing distance in miles that a car has traveled in \(t\) minutes. Find and interpret \(f^{-1} (70)\).
| \(t\text{ (minutes)}\) | 30 | 50 | 70 | 90 |
| \(f( t )\text{ (miles)}\) | 20 | 40 | 60 | 70 |
- Solution
-
The inverse function takes an output of \(f\) and returns an input for \(f\). So in the expression \(f^{-1} (70)\), 70 is an output value of the original function, representing 70 miles. The inverse will return the corresponding input of the original function \(f\), 90 minutes, so \(f^{-1} (70)=90\). The interpretation of this is that, to drive 70 miles, it took 90 minutes.
Alternatively, recall that the definition of the inverse was that if \(f(a)=b\), then \(f^{-1} (b)=a\). By this definition, if we are given \(f^{-1} (70)=a\), then we are looking for a value \(a\) so that \(f(a)=70\). In this case, we are looking for a \(t\) so that \(f(t)=70\), which is when \(t=90\).
Using Table \( \PageIndex{ 4 } \), find and interpret each of the following:
- \(f(60)\)
- \(f^{-1} (60)\)
| \(t\text{ (minutes)}\) | 30 | 50 | 60 | 70 | 90 |
| \(f( t )\text{ (miles)}\) | 20 | 40 | 50 | 60 | 70 |
Evaluating the Inverse of a Function, Given a Graph of the Original Function
We know that the domain of a function can be read by observing the horizontal extent of its graph. We find the domain of the inverse function by observing the vertical extent of the graph of the original function, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the inverse function by observing the horizontal extent of the graph of the original function, as this is the vertical extent of the inverse function. If we want to evaluate an inverse function, we find its input within its domain, which is all or part of the vertical axis of the original function’s graph.
A function \(g(x)\) is given in Figure \( \PageIndex{ 5 } \). Find \(g(3)\) and \(g^{-1} (3)\).
Figure \( \PageIndex{ 5 } \)
- Solution
-
To evaluate \(g(3)\), we find 3 on the \( x \)-axis and find the corresponding output value on the \( y \)-axis. The point \(( 3,1 )\) tells us that \(g(3)=1\).
To evaluate \(g^{-1} (3)\), recall that by definition \(g^{-1} (3)\) means the value of \( x \) for which \(g(x)=3\). By looking for the output value 3 on the vertical axis, we find the point \(( 5,3 )\) on the graph, which means \(g(5)=3\), so by definition, \(g^{-1} (3)=5\). See Figure \( \PageIndex{ 6 } \).
Figure \( \PageIndex{ 6 } \)
Using the graph in Figure \( \PageIndex{ 6 } \) to answer the following questions.
- Find \(g^{-1} (1)\).
- Estimate \(g^{-1} (4)\).
Finding Inverses of Functions Represented by Formulas
Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function is given as a formula - for example, \(y\) as a function of \(x\) - we can often find the inverse function by solving to obtain \(x\) as a function of \(y\).
Given a function represented by a formula, find the inverse.
- Make sure \(f\) is a one-to-one function.
- Solve for \(x\).
- Interchange \(x\) and \(y\).
Find a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature,\[C= \dfrac{5}{9} (F−32). \nonumber \]
- Solution
-
\[ \begin{array}{rrclcl}
& C & = & \dfrac{5}{9}(F - 32) & & \\[6pt]
\implies & \dfrac{9}{5} C & = & F - 32 & \quad & \left( \text{multiplying both sides by }\dfrac{9}{5} \right) \\[6pt]
\implies & \dfrac{9}{5} C + 32 & = & F & \quad & \left( \text{adding }32 \text{ to both sides} \right) \\[6pt]
\end{array} \nonumber \]By solving in general, we have uncovered the inverse function. If\[C=h(F)= \dfrac{5}{9} (F−32), \nonumber \]then\[F= h^{-1} (C)= \dfrac{9}{5} C+32. \nonumber \]In this case, we introduced a function \(h\) to represent the conversion because the input and output variables are descriptive, and writing \(C^{-1}\) could get confusing.
Solve for \(x\) in terms of \(y\) given \(y= \frac{1}{3} (x−5)\).
Find the inverse of the function \(f( x )= \frac{2}{x−3} +4\).
- Solution
-
\[ \begin{array}{rrclcl}
& y & = & \dfrac{2}{x - 3} + 4 & \quad & \left( \text{starting with the original equation} \right) \\[6pt]
\implies & y - 4 & = & \dfrac{2}{x - 3} & \quad & \left( \text{subtracting }4\text{ from both sides} \right) \\[6pt]
\implies & (x - 3)(y - 4) & = & 2 & \quad & \left( \text{multiplying both sides by }(x - 3) \right) \\[6pt]
\implies & x - 3 & = & \dfrac{2}{y - 4} & \quad & \left( \text{dividing both sides by }(y - 4) \right) \\[6pt]
\implies & x & = & \dfrac{2}{y - 4} + 3 & \quad & \left( \text{adding }3\text{ to both sides} \right) \\[6pt]
\end{array} \nonumber \]So \(f^{-1} ( y )= \frac{2}{y−4} +3\) or \(f^{-1} ( x )= \frac{2}{x−4} +3\).
Note that the domain and range of \(f\) in Example \( \PageIndex{ 8 } \) exclude the values \( 3 \) and \( 4 \), respectively. \(f\) and \(f^{-1}\) are equal at two points but are not the same function, as we can see by creating Table \( \PageIndex{ 5 } \).
| \(x\) | 1 | 2 | 5 | \(f^{-1} (y)\) |
| \(f(x)\) | 3 | 2 | 5 | \(y\) |
Find the inverse of the function \(f(x)=2+\sqrt{ x−4 } \).
- Solution
-
\[ \begin{array}{rrclcl}
& y & = & 2 + \sqrt{x - 4} & \quad & \left( \text{starting with the original equation} \right) \\[6pt]
\implies & y - 2 & = & \sqrt{x - 4} & \quad & \left( \text{subtracting }2\text{ from both sides} \right) \\[6pt]
\implies & \left(y - 2\right)^2 & = & x - 4 & \quad & \left( \text{squaring both sides} \right) \\[6pt]
\implies & \left(y - 2\right)^2 + 4 & = & x & \quad & \left( \text{adding }4\text{ to both sides} \right) \\[6pt]
\end{array} \nonumber \]So \(f^{-1} ( x )= ( x−2 )^2 +4\).The domain of \(f\) is \([4, \infty )\). Notice that the range of \(f\) is \([2, \infty )\), so this means that the domain of the inverse function \(f^{-1}\) is also \([2, \infty )\).
The formula we found for \(f^{-1}( x )\) in Example \( \PageIndex{ 9 } \) looks like it would be valid for all real \(x\). However, \(f^{-1}\) itself must have an inverse (namely, \(f\) ) so we have to restrict the domain of \(f^{-1}\) to \([2, \infty )\) in order to make \(f^{-1}\) a one-to-one function. This domain of \(f^{-1}\) is exactly the range of \(f\).
What is the inverse of the function \(f(x)=2−\sqrt{ x } \)? State the domains of both the function and the inverse function.
Finding Inverse Functions and Their Graphs
Now that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function \(f(x)= x^2\) restricted to the domain \([0, \infty )\), on which this function is one-to-one, and graph it as in Figure \( \PageIndex{ 7 } \).
Figure \( \PageIndex{ 7 } \): Quadratic function with domain restricted to \( [0, \infty ) \).
Restricting the domain to \([0, \infty )\) makes the function one-to-one (it will obviously pass the Horizontal Line Test), so it has an inverse on this restricted domain.
We already know that the inverse of the toolkit quadratic function is the square root function, that is, \(f^{-1} (x)=\sqrt{ x } \). What happens if we graph both \(f\) and \(f^{-1}\) on the same set of axes, using the \(x\)-axis for the input to both \(f\) and \(f^{-1} \)?
We notice a distinct relationship: The graph of \( f^{-1}(x) \) is the graph of \(f(x)\) reflected about the diagonal line \(y=x\), which we will call the identity line, shown in Figure \( \PageIndex{ 8 } \).
Figure \( \PageIndex{ 8 } \): Square and square-root functions on the non-negative domain
This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.
Given the graph of \(f(x)\) in Figure \( \PageIndex{ 9 } \), sketch a graph of \(f^{-1} (x)\).
Figure \( \PageIndex{ 9 } \)
- Solution
-
This is a one-to-one function, so we will be able to sketch an inverse. Note that the graph shown has an apparent domain of \(( 0, \infty )\) and range of \(( − \infty , \infty )\), so the inverse will have a domain of \(( − \infty , \infty )\) and range of \(( 0, \infty )\).
If we reflect this graph over the line \(y=x\), the point \(( 1,0 )\) reflects to \(( 0,1 )\) and the point \(( 4,2 )\) reflects to \(( 2,4 )\). Sketching the inverse on the same axes as the original graph gives Figure \( \PageIndex{ 10 } \).
Figure \( \PageIndex{ 10 } \): The function and its inverse, showing reflection about the identity line
Draw graphs of the functions \(f\) and \(f^{-1}\) from Example \( \PageIndex{ 8 } \).
Is there any function that is equal to its own inverse?
Yes. If \(f= f^{-1} \), then \(f( f( x ) )=x\), and we can think of several functions that have this property. The identity function does, and so does the reciprocal function, because\[\dfrac{1}{\frac{1}{x}} =x. \nonumber \]Any function \(f( x )=c−x\), where \(c\) is a constant, is also equal to its own inverse.
Access these online resources for additional instruction and practice with inverse functions.


4.5: Systems of Inequality
-
- Last updated
- Save as PDF
- Jennifer Hasty, David G. Lewis, & Marjorie M. Snipes
- OpenStax
Learning Objectives
By the end of this section, you will be able to:
- Explain the meanings of the terms racism, Whiteness, and White supremacy.
- Differentiate between economic, social, and cultural capital in relation to class or social mobility.
- Explain the relationship between capitalism and social inequalities.
- Describe gender relations, patriarchy, and oppression.
Many introductory anthropological texts will examine how types of social stratification align with modes of production. This text has something of a different focus, critically considering what it means for some lives to matter more or less than others. This section looks at how modern modes of production create systems of social inequalities such as racism, classism, and sexism.
Race and Racism
Racism is best understood as power intertwined with racial prejudice. Racism can be perpetuated through interpersonal, institutional, and systemic practices. Anthropologists Alan Goodman, Yolanda Moses, and Joseph Jones define racism in Race: Are We So Different? (2020) as the use of race to establish and justify a social hierarchy and system of power that privileges and advances certain individuals or groups of people, usually at the expense of others. Many individuals understand interpersonal examples of racism, but what are institutional or systemic forms of racism? To explore this question, this section will discuss the history of race and its social construction.
What Is Anthropology? discussed the fact that race is a social construct. Where did the social construct of race originate? Johann Blumenbach, a German physician and anthropologist, was influential in establishing existing racial categories. Working in the field of craniometry, a now debunked pseudoscience that studied human head shape and brain size, Blumenbach proposed five racial categories to divide humans in the late 1700s: “Caucasian” for White people, “Mongolian” for Asians, “Malayan” for Brown people, “Ethiopian” for Black people, and “American” for Indigenous people of the Americas (Goodman, Moses, and Jones 2020, 30).
Blumenbach intentionally made these categories hierarchical and put White people at the top of this hierarchy. In many ways, the remnants of this hierarchy still exist today. For instance, have you ever seen the term Caucasian on a form asking about race? Why does this term still exist? Many other labels from the classifications Blumenbach created have been challenged, but Caucasian is still used in both scientific and popular usage. Anthropologist Carol Mukhopadhyay (2008) argues that this term’s continued usage conveys a false scientific authority of Whiteness.
Black anthropologists, including Williams S. Willis Jr. (1972) and others, have pointed out many racist undertones throughout anthropology’s history of studying the “other.” Anthropology began as the practice of White anthropologists studying the non-White other, which was rooted in an inherently unequal perspective. The White anthropologists’ beliefs were considered the “norm,” and people they studied were considered outside of the norm. In contrast, many of the first Black anthropologists trained in the United States were involved in activism, advocacy, public service, and social justice. These Black pioneers in anthropology were committed to fighting racism and instigating social change, focuses that were reflected in their scholarship and how they approached anthropology (Harrison and Harrison 1999). In “Reflections on Anthropology and the Black Experience,” St. Clair Drake, discussing why some Black scholars became anthropologists, said, “A few of us chose careers in anthropology forty to forty-five years ago because we believed the discipline had relevance to the liberation of black people from the devastating consequences of over four centuries of white racism” (1978, 86).
In 1941, anthropologists Allison Davis, Burleigh Gardner, and Mary Gardner argued that the United States had a racial caste system. Caste is a system of social inequality based on an individual’s circumstances of birth, wherein people are not allowed to move out of the social group into which they are born. Davis, Gardner, and Gardner observe that racism is a powerful force in American society that produces inequitable social relations that seem permanent but vary regionally and are subject to change over time. They argue that political, social, and economic structures all maintain that caste system, often in violent and coercive ways (Davis, Gardner, and Gardner 1941).
A number of scholars have also examined White racial identity; these “Whiteness studies” show that the racial category of White has been defined in different ways throughout US history. For instance, certain ethnicities in American history were not originally considered White but became included in the White identity over time. Whiteness is usually based on the maintenance or pursuit of power and proximity to power. Historian Nell Irvin Painter’s book The History of White People (2010) provides a detailed history of European civilization, race, and the frequent worshipping of Whiteness and explains that the concept of one White race is a recent invention.
White privilege is conceptualized as the ways in which White people have been given advantages at the expense of other populations. In Peggy McIntosh’s classic article “White Privilege: Unpacking the Invisible Knapsack” (1989), she compares White privilege to an invisible weightless knapsack that comes with special provisions or advantages. According to McIntosh (who identifies as White), these advantages—or even just lack of obstacles—include not having to think about one’s race all the time, knowing that one will probably be represented wherever they go, and not worrying about having to speak for all the people of one’s racial group, among many other examples. Thus, White privilege is the experience of one’s Whiteness as the standard.
White privilege is often linked to the cultural concept of White supremacy , which is the idea that White people are a superior race and should dominate society at the expense of other, historically oppressed groups. People often think of White supremacy as extremist behavior, but White supremacy can actually be seen in many examples of systemic social inequalities. Ideologies of the Ku Klux Klan and neo-Nazis are examples of overt White supremacy that many people acknowledge as being racist. However, there are many covert examples of White supremacy that are problematic and racist but are overlooked.
The concept of White supremacy is a contentious one in modern media and politics. You may have come across an image like the one in Figure 9.7 explaining different types of White supremacy. Although the examples in the diagram labeled “Overt” can be agreed on as socially unacceptable by most people in American society, the examples in the “Covert” section are often explained on an individual level instead of as a symptom of racism. For instance, the school-to-prison pipeline can often be explained as the consequence of individuals who do not obey the rules instead of a consequence of underfunded schools and racist policies.
The avoidance of talking about race, or racial refusal , can be understood as a silent form of racism. Anthropologist Dána-Ain Davis, in her ethnography Reproductive Injustice: Racism, Pregnancy, and Premature Birth (2019), writes that not acknowledging race in certain contexts can perpetuate inequalities. For her study of Black women who give birth to premature infants, Davis interviewed Black mothers and their partners; NICU (neonatal intensive care unit) staff, including nurses and doctors; birth workers; and March of Dimes administrators. In her research, Davis found that many doctors refused to discuss race and consequently ignored how racism is connected to disparities in health, premature birth, and medical treatment. Instead, discussions of premature birth disparities centered on class, despite the fact that Davis interviewed professional Black women who were college educated. Davis argues that racial disparities and medical racism perpetuated by systemic and structural racism cannot be addressed in healthcare settings if healthcare workers do not discuss race. This racial refusal has a historical precedence in the United States, where history and how that history has affected people’s lives is routinely omitted (Davis 2019, 88).
Finally, microaggressions are everyday instances of racism, homophobia, sexism, ableism, and other isms that are observed in the world as thinly veiled insults directed toward individuals from historically excluded groups.  Microaggressions include verbal and nonverbal snubs and insults that communicate hostile, derogatory, or negative messages to individuals based solely on their identification with a marginalized group. For example, one of the coauthors of this chapter, Saira Mehmood, identifies as a Muslim woman of South Asian descent, born in New Orleans. Saira is often asked, “Where are you from?” When she answers, “New Orleans,” the next question is often “Where are you really from?” This type of microaggression denies Saira’s agency as an American.
Class
Class refers to a group of people with the same socioeconomic status and proximity to power. In a class-based system, status stems from wealth and one’s proximity to the power that wealth builds. Economically, class systems are most often associated with the capitalist mode of production. People in the United States often think of the term middle class when considering class systems.
Capitalism — the economic mode of production based around markets, ownership of land and resources, and wage labor—has produced classes that are grounded in the acceptance of the idea that earned wealth or status is the basis for social hierarchy within a nation. In capitalist nations, a person’s status in society directly relates to the amount of money they have acquired or the position they have achieved in their career. Class-based systems often emphasize social inequalities because of the hegemonic idea that relation to capital determines a person’s value in society. For instance, Bill Gates is looked up to for his status as a billionaire, while those who work in fast food are often seen as not deserving of a living wage. This system of inequality, especially in the United States, is tied to the idea of meritocracy, with those at the top of the class system assumed to have worked hardest or to be most deserving of high-level positions and those at the bottom assumed to be personally at fault for their lack of wealth.
Capitalism includes the concept of social mobility , or the ability of an individual to move up into higher and thus more powerful classes merely by working hard. Social mobility is the basis for the “American Dream,” the idea that poor Americans can attain a higher class. On the other hand, anthropologist Katherine S. Newman has done decades of research on downward social mobility , or the ongoing loss of capital and ensuing loss of social status. Newman (1999) found that in the last decades of the 20th century, divorce, emigration, company downsizing, and technological advancement left many middle-class individuals struggling to maintain their class (also see Gans 2009). Furthermore, the 2008 recession and the economic crash experienced due to the COVID-19 pandemic led to downward social mobility for millions.
In addition to class, the United States also uses the concept of “collar.” White-collar jobs are assumed to require higher education, involve less manual labor, and pay more, while blue-collar jobs are considered less skilled, more manual, and lower paying. However, Forbes magazine found that there are many “blue collar” jobs (e.g., plumbers and electricians) that have higher earnings than many “white collar” jobs (such as entry- or mid-level finance), yet they carry lower status within US social hierarchy. What distinguishes white-collar from blue-collar jobs if it isn’t just about how much money they make? German social scientist Max Weber argued that there were considerably more than two classes that determined the social inequalities and conflicts among people in capitalist societies. In his seminal essay “The Distribution of Power with the Community: Classes, Stände, Parties” (2010), originally published in German in 1921, Weber argues that there are multiple, overlapping systems from which to gain power and links social stratification to three components: socioeconomic status, prestige, and political party connections.
Power, in capitalist and class societies, often stems from capital, which is wealth in the form of money or other assets. Economic capital is monetary but is not the only form of capital. French sociologist Pierre Bourdieu distinguished between various forms of capital: economic, social, cultural, and symbolic. Bourdieu defined social capital as the nonmonetary resources people use to gain social status, such as mutual acquaintances, shared cultural knowledge, or shared experiences. Social capital can also determine one’s power. Cultural capital refers to the competencies, skills, and qualifications that people acquire that create cultural authority; in an institutionalized form, this takes the form of educational attainment. Symbolic capital , or the resources available to an individual because of honor, prestige, or recognition, is tied to economic, social, and cultural capital. For instance, successful athletes often have symbolic capital, and this type of capital can increase their social capital and economic capital with endorsements from corporations and other opportunities. However, athletes can also lose their symbolic capital when a scandal or controversy involving them is uncovered, resulting in them losing their endorsements and contracts, which in turn affects their economic and social capital.
A good example of how individuals utilize social capital in the United States is the networking that exists in top-tier schools. In Pedigree: How Elite Students Get Elite Jobs (2016), sociologist Lauren Rivera utilizes participant observation to show how top-tier investment banks, consulting firms, and law firms decide who gets hired and who doesn’t, drawing on analysis of social and cultural capital in the American class system. Often, interviewers from elite firms use the phrase “not a good fit” when deciding not to hire someone in order to skirt around potential accusations of discriminatory intent. Riviera concludes that if a candidate is not from a top-tier school, the only way for them to get hired by such a firm is to have some other social capital connection vouch for their abilities.
When those with symbolic capital use their power against those with less power in order to change their actions, they are exercising symbolic violence. Symbolic violence is a type of nonphysical violence manifested in power differentials between social groups (e.g., upper class and lower class). For Bourdieu, symbolic violence reinforces ideologies that legitimize and naturalize the status quo. In many instances, symbolic violence reinforces social inequalities. This is perhaps most evident in the language used when referring to other groups. During the long history of migrations toward the US-Mexico border, symbolic violence has been used linguistically by English speakers to refer to migrants in terms that alienate them and set them outside of a common human identity. Labels such as “illegals,” “illegal aliens,” and “undocumented workers” are applied across cultures, defining families and individuals by a single dimension. Linguistic slurs are especially associated with symbolic violence. When human beings are represented in such simple and stark terms, it can become more socially acceptable to oppress them and see them as undeserving of empathy and respect.
Capitalism and class systems can also be analyzed in terms of race. Initially popularized by political science and Black studies scholar Cedric J. Robinson in Black Marxism: The Making of the Black Radical Tradition (1983), racial capitalism is the process through which the key aspects of capitalism (credit/debit, production/surplus, capitalist/worker, developed/underdeveloped, etc.) become articulated through existing relations of racial inequalities. In Robinson’s framework, capitalism is racial not because of some conspiracy to divide workers or to justify slavery but because racialism had already spread through Western feudal society when capitalism developed. Racial capitalism can clearly be seen in the slave trade and colonialism.  Hartman describes this as “the afterlife of slavery—skewed life chances, limited access to health and education, premature death, incarceration, and impoverishment” (2007, 6). Slavery was a racialized system of capitalism, one that continues to exploit others to the present day.
Class systems emphasize social inequalities because for some people to have money and power, those individuals must exploit and oppress other groups. Capitalism and class societies are often supported by the ideas that those with power earned that power and those without it have individual moral failings instead of acknowledging that the structure of capitalism, which necessitates a working class, generates inequalities.
Gender and Patriarchy
Although there is a detailed exploration of gender, patriarchy, and power in Gender and Sexuality , this chapter will discuss how gender is tied to social inequalities. Anthropologists have studied how gender relations play a big part in experiences of inequality. Gender relations can interact with various other powerful cultural institutions to further oppress individuals.
An important concept to grasp when seeking to understand gender and power is patriarchy , a system of social inequality based on gender in which power is assumed to be in the hands of men and characteristics associated with femininity are less valued. Patriarchy is related to male lineages and contexts in which men hold more political, social, and economic power or prestige. Recently, the claim that patriarchy remains a powerful force has been challenged by some social commentators, who argue that this system of oppression does not exist in modern society and that women and men experience equal opportunities in terms of employment, rights, and salary. Many anthropologists and other social scientists challenge this claim, pointing out ways in which patriarchy still impacts women’s lives.
Many anthropologists have made connections between gender and patriarchy, poverty, and race. In her fieldwork in the poor, mostly Black midwestern suburb of “Meadow View,” sociologist Sharon Hicks-Bartlett (2000) observed a particular type of oppression experienced by local women. Women living in poverty were relied upon and expected to keep their families together. Hicks-Bartlett described women tasked with managing low-wage, part-time work in a place where public systems of care and assistance, or even buses, were largely unavailable.
The interpersonal and even internalized forces of patriarchy and power can also make women “compete to lose,” meaning they will deliberately not succeed at some things in order to gain social capital among their peers. For instance, anthropologist Signithia Fordham, (2013) who spent two years studying the interactions of Black teenage girls in a predominantly White high school (which she aptly named “Underground Railroad High School”), found that the girls in this middle-class high school downplayed their achievements in order to fit in with peer groups and friends. Academic success was sometimes experienced as a social hindrance for those whose goals were family and children.
Profiles in Anthropology
Dr. William S. Willis Jr. (1921–1983)
Personal History: Dr. William S. Willis Jr. was a Black intellectual, anthropologist, historian, and anti-racist scholar of the 20th century. He was born in Waco, Texas, but his family moved to Dallas because of threats from the Waco Ku Klux Klan. After graduating from Howard University as a history major, Willis volunteered for service with the US Coast Guard. Eventually, he began his graduate studies in anthropology at Columbia University, drawn to the program by the scientific anti-racism of the Boasian tradition.
Area of Anthropology: As a graduate student, Willis wanted to study Black culture and Black relations at home and abroad, but he was not able to do so because of the dominance of the study of Native Americans in American anthropology at the time. Nevertheless, Willis remained convinced of the importance of the historical approach in anthropology and of studying cultural change through time, considerations that were largely ignored by other theoretical frameworks popular in anthropology at the time.
Importance of His Work: Willis became the first Black faculty member at Southern Methodist University (SMU). While he was popular as a professor in the Department of Sociology and Anthropology at SMU, he faced numerous hurdles. He received the least pay and has said that he felt like he was the “workhorse of the department” (quoted in Harrison and Harrison 1999, 253), teaching the greatest number of new courses. Despite being promoted to associate professor with tenure, Willis resigned from SMU in 1972, citing the covert and overt racism he experienced in the anthropology department.
His 1972 article “Skeletons in the Anthropological Closet,” published in Reinventing Anthropology , declared that anthropology’s claim of being the “science of man” was delusional and asserted that anthropology’s virtual silence on the domination and exploitation of people of color at home and abroad, living outside the boundaries of White societies, was not consistent with the field’s tradition of scientific anti-racism. Willis argued that anthropology was organized around the needs of White people and that most White anthropologists did not see people of color as real human beings.


2.2.2: The pressure-volume relationship
-
- Last updated
- Save as PDF
Pressure-volume relationship
Consider a gas in a cylinder with a piston pushing it down due to massive objects placed on it and due to outside air pressure, as illustrated in Fig. 7.2.1, The gas molecules strike the piston surface, applying pressure upward equal to the downward pressure applied by the piston that keeps the piston stationary. Increasing the pressure on the piston, e.g., by adding more weight to it, causes the piston to move down, reducing the gas volume. The gas molecules have less distance to travel before striking the piston surface which increases the collision frequency and causes the gas pressure to increase until becomes equal to the outside pressure.
When a change in one parameter causes a change in another, the parameters are related. When an increase in one parameter causes a decrease in another, the two are inversely proportional to each other. Robert Boyle studied the quantitative relationship between the volume and pressure of the gas, keeping the quantity of gas and the temperature constant. The research concluded in a law called Boyle’s law, which states that:
Boyle's law
The volume of a gas is inversely proportional to the pressure of the gas provided the temperature and the amount of the gas remain constant.
Fig. 7.2.2 illustrates the results of Boyle’s experiment. The mathematical form of Boyle’s law is: \[V \propto \frac{1}{P}\nonumber\], or \[V=\frac{\mathrm{k}}{P}\nonumber\], or \[PV=\mathrm{k}\nonumber\], where k is a consant. The graph of volume versus pressure is curvilinear, but the graph of volume versus the reciprocal of pressure is a linear graph showing the inverse proportionality between the volume and the pressure. Since the product PV is a constant, it implies that: \[P_{1} V_{1}=P_{2} V_{2}={k}\nonumber\], i.e., a product of initial pressure ( P 1 ) and initial volume ( V 1 ) is equal to the product of final pressure ( P 2 ) and final volume ( V 2 ) of gas provided the quantity of the gas and temperature does not change.
Example \(\PageIndex{1}\)
The pressure of a 1.32 L sample of SO 2 gas at 0.532 atm is increased to 1.231 atm. Calculate the new volume of the gas if the temperature and the quantity of the gas remain the same?
Solution
Given: P 1 = 0.532 atm, P 2 = 1.231 atm, V 1 = 1.32 L V 2 = ?
Formula: \(P_{1} V_{1}=P_{2} V_{2}={k}\), rearrange to isolate the desired variable: \(V_{2}=\frac{P_{1} V_{1}}{P_{2}}\).
Plug in the values in the rearranged formula and calculate: \(V_{2}=\frac{0.532 \mathrm{~atm} \times 1.32 \mathrm{~L}}{1.231 \mathrm{~atm}}=0.570 \mathrm{~L}\)
Example \(\PageIndex{2}\)
An oxygen tank holds 20.0 L of oxygen at a pressure of 10.0 atm. What is the final pressure when the gas is released and occupies a volume of 200 L?
Solution
Given: V 1 = 20.0 L, V 2 = 200 L, P 1 = 10.0 atm P 2 = ?
Formula: \(P_{1} V_{1}=P_{2} V_{2}={k}\), rearrange to isolate the desired variable: \(P_{2}=\frac{P_{1} V_{1}}{V_{2}}\).
Plug in the values in the rearranged formula and calculate: \(P_{2}=\frac{10.0 \mathrm{~atm} \times 20.0 \mathrm{~L}}{200 \mathrm{~L}}=1.00 \mathrm{~atm}\).
Breathing process
Boyle’s law explains the mechanism of the breathing process. Lungs are elastic structures like balloons placed in the thoracic cavity, as illustrated in Fig. 7.2.3. The diaphragm muscle makes a flexible floor and ribs surround the cavity.
Inhalation
The inhalation or the inspiration process starts when the diaphragm contract and move down and the rib muscles contract, expanding the thoracic cavity. Volume increases, the air pressure decreases inside the inside thoracic cavity and the atmospheric air flows into the lungs until the pressure in the lungs is equal to the outside pressure.
Exhalation
The exhalation or the expiration process starts when the diaphragm expands and moves upwards, and the rib muscles relax, contracting the thoracic cavity. Volume decreases and the air pressure increases inside the thoracic cavity that pumps the air out of the lungs into the atmosphere.


5.10: Introduction to Barriers to Individual Decision Making and Styles of Decision Making
What you’ll learn to do: describe the barriers to individual decision making and common styles of decision making
Making decisions is easy. Making the right decision is hard. When making decisions, you will face many barriers, including the quality of information you have, the amount of time allowed, and several cognitive biases that will influence your decisions. In addition to these barriers, we’ll also look at some common styles of decision making, including satisficing, optimizing, intuitive, rational, combinatorial, and positional.
Contributors and Attributions
CC licensed content, Original
- Introduction: Barriers to Individual Decision Making and Common Styles of Decision Making . Authored by : Aaron Spencer and Lumen Learning. . License : CC BY: Attribution


8.17D: Chytridiomycota - The Chytrids
- Describe the ecology and reproduction of chytrids
Chytridiomycota: The Chytrids
The kingdom Fungi contains five major phyla, which were established according to their mode of sexual reproduction or use of molecular data. The Phylum Chytridiomycota (chytrids) is one of the five true phyla of fungi. There is only one class in the Phylum Chytridiomycota, the Chytridiomycetes. The chytrids are the simplest and most primitive Eumycota, or true fungi. The evolutionary record shows that the first, recognizable chytrids appeared during the late pre-Cambrian period, more than 500 million years ago. Like all fungi, chytrids have chitin in their cell walls, but one group of chytrids has both cellulose and chitin in the cell wall. Most chytrids are unicellular; a few form multicellular organisms and hyphae, which have no septa between cells (coenocytic). They reproduce both sexually and asexually; the asexual spores are called diploid zoospores. Their gametes are the only fungal cells known to have a flagellum.
The ecological habitat and cell structure of chytrids have much in common with protists. Chytrids usually live in aquatic environments, although some species live on land. Some species thrive as parasites on plants, insects, or amphibians, while others are saprobes. Some chytrids cause diseases in many species of amphibians, resulting in species decline and extinction. An example of a harmful parasitic chytrid is Batrachochytrium dendrobatidis , which is known to cause skin disease. Another chytrid species, Allomyces , is well characterized as an experimental organism. Its reproductive cycle includes both asexual and sexual phases. Allomyces produces diploid or haploid flagellated zoospores in a sporangium.
Key Points
- The first recognizable chytrids appeared more than 500 million years ago during the late pre-Cambrian period.
- Like protists, chytrids usually live in aquatic environments, but some species live on land.
- Some chytrids are saprobes while others are parasites that may be harmful to amphibians and other animals.
- Chytrids reproduce both sexually and asexually, which leads to the production of zoospores.
- Chytrids have chitin in their cell walls; one unique group also has cellulose along with chitin.
- Chytrids are mostly unicellular, but multicellular organisms do exist.
Key Terms
- chytridiomycete : an organism of the phylum Chytridiomycota
- zoospore : an asexual spore of some algae and fungi
- flagellum : a flagellum is a lash-like appendage that protrudes from the cell body of certain prokaryotic and eukaryotic cells
- coenocytic : a multinucleate cell that can result from multiple nuclear divisions without their accompanying cytokinesis