1. Section 11.2
Inverse Functions

2. Objectives Determine whether a function is a one-to-one function
Use the horizontal line test to determine whether a function is one-to-one
Find the equation of the inverse of a function
Find the composition of a function and its inverse
Graph a function and its inverse

3. Objective 1: Determine Whether a Function Is a One-to-One Function
In figure (a) below, the arrow diagram defines a function ƒ. If we reverse the arrows as shown in figure (b), we obtain a new correspondence where the range of ƒ becomes the domain of the new correspondence, and the domain of ƒ becomes the range. The new correspondence is a function because to each member of the domain, there corresponds exactly one member of the range. We call this new correspondence the inverse of ƒ, or ƒ inverse.

4. Objective 1: Determine Whether a Function Is a One-to-One Function
What must be true of an original function to guarantee that the reversing process produces a function? The answer is: the original function must be one-to-one. In a function, each input determines exactly one output. For some functions, different inputs determine different outputs, as in figure (a). For other functions, different inputs might determine the same output, as in figure (b). When a function has the property that different inputs determine different outputs, as in figure (a), we say the function is one-to-one.
A function is called a one-to-one function if different inputs determine different outputs.

5. EXAMPLE 1
Determine whether each function is one-to-one.
a. ƒ(x) = x2 b. ƒ(x) = x3
Strategy We will determine whether different inputs have different outputs.
Why If different inputs have different outputs, the function is one-to-one. If different inputs have the same output, the function is not one-to-one.

6. EXAMPLE 1
Determine whether each function is one-to-one.
a. ƒ(x) = x2 b. ƒ(x) = x3
Solution
a. Since two different inputs, –3 and 3, have the same output, 9, ƒ(x) = x2 is not a one-to-one function.
b. Since different numbers have different cubes, each input of ƒ(x) = x3
determines a different output. This function is one-to-one.

7. Objective 2: Use the Horizontal Line Test to Determine Whether a Function Is One-to-One
Horizontal Line Test: A function is one-to-one if each horizontal line that intersects its graph does so exactly once.
To determine whether a function is one-to-one, it is often easier to view its graph rather than its defining equation. If two (or more) points on the graph of a function have the same y-coordinate, the function is not one-to-one.

8. EXAMPLE 2
Use the horizontal line test to determine whether the following graphs of functions represent one-to-one functions.
Strategy We will draw horizontal lines through the graph of the function and see how many times each line intersects the graph.
Why If each horizontal line intersects the graph of the function exactly once, the graph represents a one-to-one function. If any horizontal line intersects the graph of the function more than once, the graph does not represent a one-to-one function.

9. EXAMPLE 2
Use the horizontal line test to determine whether the following graphs of functions represent one-to-one functions.
Solution
a. Because every horizontal line that intersects the graph of in figure (a) does so exactly once, the graph represents a one-to-one function. We simply say, the function is one-to-one.
b. Refer to figure (b). Because we can draw a horizontal line that intersects the graph of ƒ(x) = x2 – 4 twice, the graph does not represent a one-to-one function. We simply say,
the function ƒ(x) = x2 – 4 is not one-to-one.

10. EXAMPLE 2
Use the horizontal line test to determine whether the following graphs of functions represent one-to-one functions.
Solution
c. Because every horizontal line that intersects the
graph of in figure (c) does so exactly once, the graph represents a one-to-one function.
We simply say, the function is one-to-one.

11. Objective 3: Find the Equation of the Inverse of a Function
The Inverse of a Function: If ƒ is a one-to-one function consisting of ordered pairs of the form (x, y), the inverse of ƒ, denoted ƒ-1, is the one-to-one function consisting of all ordered pairs of the form (y, x).
Finding the Equation of an Inverse of a Function:
If a function is one-to-one, we find its inverse as follows:
If the function is written using function notation, replace ƒ(x) with y.
Interchange the variables x and y.
Solve the resulting equation for y.
Substitute ƒ-1(x) for y.

12. EXAMPLE 3
Determine whether each function is one-to-one. If so, find the equation of its inverse.
a. ƒ(x) = 4x b. ƒ(x) = x3
Strategy We will determine whether each function is one-to-one. If it is, we can find the equation of its inverse by replacing ƒ(x) with y, interchanging x and y, and solving for y.
Why The reason for interchanging the variables is this: If a one-to-one function takes an input x into an output y, by definition, its inverse function has the reverse effect.

13. EXAMPLE 3
Determine whether each function is one-to-one. If so, find the equation of its inverse.
a. ƒ(x) = 4x b. ƒ(x) = x3
Solution
a. We recognize ƒ(x) = 4x + 2 as a linear function whose graph is a straight line with slope 4 and y-intercept (0, 2). Since such a graph would pass the horizontal line test, we conclude that ƒ is one-to-one. To find the inverse, we proceed as follows:
Function f multiplies all inputs by 4 and then adds 2.
Function f-1 subtracts 2 from each input and then divides by 4.

14. EXAMPLE 3
Determine whether each function is one-to-one. If so, find the equation of its inverse.
a. ƒ(x) = 4x b. ƒ(x) = x3
Solution
b. The graph of ƒ(x) = x3 is shown on the right. Since such a graph would pass the horizontal line test, we conclude that ƒ is a one-to-one function. To find its inverse, we proceed as follows:
Function f cubes all inputs.
Function f-1 finds the cube root of each input.

15. Objective 4: Find the Composition of a Function and Its Inverse
For any one-to-one function ƒ and its inverse ƒ-1,
We can use this property to determine whether two functions are inverses.

16. EXAMPLE 4
Strategy We will find the composition of ƒ(x) and ƒ-1(x) in both directions and show that the result is x.
Why Only when the result of the composition is x in both directions are the functions inverses.

17. EXAMPLE 4
Solution

18. Objective 5: Graph a Function and Its Inverse
If a point (a, b) is on the graph of function ƒ, it follows that the point (b, a) is on the graph of ƒ-1, and vice versa. There is a geometric relationship between a pair of points whose coordinates are interchanged. For example, in the graph, we see that the line segment between (1, 3) and (3, 1) is perpendicular to and cut in half by the line y = x. We say that (1, 3) and (3, 1) are mirror images of each other with respect to y = x. Since each point on the graph of ƒ-1 is a mirror image of a point on the graph of ƒ, and vice versa, the graphs of ƒ and ƒ-1 must be mirror images of each other with respect to y = x.

19. Find the equation of the inverse of
. Then graph ƒ and its inverse on one coordinate system.
EXAMPLE 5
Strategy We will determine whether the function has an inverse. If so, we will replace ƒ(x) with y, interchange x and y, and solve for y to obtain the equation of the inverse.
Why The reason for interchanging the variables is this: If a one-to-one function takes an input x into an output y, by definition, its inverse function has the reverse effect.

20. Find the equation of the inverse of
. Then graph ƒ and its inverse on one coordinate system.
EXAMPLE 5
Solution
Since is a linear function, it is one-to-one and has an inverse. To find the inverse function, we replace ƒ(x) with y, and interchange x and y to obtain
Then we solve for y to get

21. Find the equation of the inverse of
. Then graph ƒ and its inverse on one coordinate system.
EXAMPLE 5
Solution
To graph ƒ and ƒ-1, we construct tables of values and plot points. Because the functions are inverses of each other, their graphs are mirror images about the line y = x.