A function is a rule that assigns to each input number exactly one output number. The set of input numbers to which the rule applies is called the domain of the function. The set of all output numbers is called the range.

A variable that represents input numbers for a function is called an independent variable. A variable that represents output numbers is called a dependent variable.

Output numbers such as \emph{f}(-4) are called function values; they are in the range of \emph{f}.

Lets be specific about the domain of a function. Unless otherwise stated, the domain consists of all real numbers for which the equation makes sense and gives function values that are real numbers.


  • We cannot divide by zero
  • \sqrt{x} where x < 0 does not give a real number (imaginary number)

Types of functions

Any function in the form \emph{f}(x) = c, where c is a constant, is called a constant function.

A constant function belongs to a broader class of functions, called polynomial functions. in general a function of the form \emph{f}(x) = c_{n}x^{n} + c_{n-1}x^{n-1} + c_{n}x^{n} + \ldots + c_{1}x + c_{0} The number n is called the degree of a function, and c_{n} is the leading coefficient. Note that a nonzero constant function, such as \emph{f}(x) = 5 [which can be written as \emph{f}(x) = 5x^{0}], is a polynomial function of degree 0.

A function that is the quotient of polynomial functions is called a rational function. \emph{f}(x) = \frac{x^{2}-6x}{x + 5}

Sometimes more than one equation is needed to define a function.

\emph{f}(x) = \left\{ \begin{array}{rcr} 1, & \text{if} & -1 \leq s < 1, \\ 0, & \text{if} & 1 \leq s \leq 2, \\ s - 3, & \text{if} & 2 < s \leq 3. \end{array} \right.

This is called a compound function.

The function \emph{f}(x) = \left | x \right | is called the absolute function. Recall that that the absolute value, of a real number x is denoted \left | x \right | and is defined by,

\left | x \right | = \left\{ \begin{array}{rcr} x, & \text{if} & x \geq 0, \\ -x, & \text{if} & x < 0. \end{array} \right.


The graph of a function \emph{f} is simply the graph of the function \emph{y} = \emph{f}(x). It consists of all points (x, y).

In general, the domain consists of all x-values that are included in the graph, and the range is all y-values that are included.

f(x) = x^2 + 2x + 3

f(x) = x^2 + 2x + 3

It is clear that the domain of this function is all real numbers and the range is all reals \geq -4

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