We Promise to Make your Math Frustrations Go Away!


Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

Inverse Functions

Definition 1.1. We will define another special type of composition function,
called the inverse function. A function g is called the inverse function of f


Be careful to notice that for functions to be inverses, both criteria must be

Example 1.2. Let f (x) = 3x + 1 and  . Notice that f and g are
inverse functions:

Since f (g (x)) = x = g (f (x)) for all x, they must be inverse functions.

Example 1.3. Let and . Notice that f and g are inverse

Since f (g (x)) = g (f (x)) for all x, they must be inverse functions.

Example 1.4. Let , and  What happens to the criteria
for inverses?

Thus, f and g are not inverse function, if we are using all of R as our domain.
The are inverses if we only use [0,∞) as our domain.

Item 1.5. Really, all an inverse function does, is answer the question, “what x
value will give me a y value of --?” Inverses switch the input and the output of a
function, since we HAVE the y value, and WANT the x value, as opposed to the
usual case. Since this switching of input and output is what inverse functions
do, we can use this tool to actually FIND the inverse of a function.

Example 1.6. Find the inverse function for f (x) = 7x + 2. To do so, we will
make use of the same procedure we used to find the range of a function: we will
solve for x in terms of y:

Since inverse functions switch the input and output, change y and x:

is the inverse of f.
Example 1.7. Use the same procedure to find the inverses for ,
and .

Item 1.8. Notice that . Use f (x) = 2x + 1 as a counterexample.

Item 1.9. If there the ordered pair (4, 5) is on the graph of f (x), then the
ordered pair (5, 4) is on the graph of f -1(x), since inverses switch input and
output. If we keep plotting points on f and on f -1, we’ll notice that f -1 is just
a reflection of f over the line y − x. As examples, plot the following graphs:

on the right half-plane, [0,∞)

Definition 1.10. A function is said to be one to one if every y value is the
image of exactly one x value

Item 1.11. If we reflect a vertical line, which will tell us if a graph is a function
(remember the vertical line test gives us a function if the line hits the graph
at only one point), about the line y = x, since that’s what inverses do, we get
a horizontal line. So if an inverse is a function, then reflection across the line
y = x gives us a horizontal line crossing the original function exactly once.
For examples, graph y = x2, which fails the test, and y = x3, which passes the
horizontal line test.

Example 1.12. Show algebraically that y = 7x + 2 has an inverse. Assume
that there are two values, a and b that have the same y value:

So if two numbers give us the same y value, they have to be the same. Thus,
7x + 2 has an inverse.

Example 1.13. Show algebraically that y = x2 +2 has no inverse. Notice that
f(2) = 6 = f(−2). So there are two x values that give the same y values, and
x2 + 2 has no inverse.

Item 1.14. Please do the following problems on pages 238ff. to turn in:

1, 5, 9, 13, 23, 25, 27, 29, 31, 33