1. 5.2 Inverse Functions and Their Representations
Calculate inverse operations
Identify one-to-one functions
Find inverse functions symbolically
Use other representations to find inverse functions

2. Example: Finding inverse actions and operations
For each of the following, state the inverse actions or operations.
Put on a coat and go outside.
Subtract 5 from x and divide the result by 2.
Solution
Reverse order, apply inverse actions
Come inside and take off coat.
Multiply x by 2 and add 5.

3. Inverse Functions (1 of 2)
Gallons to Pints f(x) = 8x x 1 2 3 4
f(x) 8 16 24 32 x 8 16 24 32
f −1(x) 1 2 3 4
Function g performs the inverse operation of f.

4. Inverse Functions (2 of 2)
Inputs and outputs (domains and ranges) are interchanged for inverse functions.

5. Composition with Inverse Functions
The composition of a function with its inverse using input x produces output x.

6. Determine If a Function Has an Inverse
If different inputs of a function f produce the same output, then an inverse function of f does not exist. However, if different inputs always produce different outputs, f is a one- to-one function. Every one-to-one function has an inverse function.

7. One-to-One Function
A function f is a one-to-one function if, for elements c and d in the domain of f, c ≠ d implies f(c) ≠ f(d). That is, different inputs always result in different outputs.

8. Example: Determining if a function is one-to-one graphically (1 of 3)
Use each graph to determine if f is one-to-one and if f has an inverse function.

9. Example: Determining if a function is one-to-one graphically (2 of 3)
Solution a. The horizontal line y = 2 intersects the graph of f at (−1, 2), (1, 2) and (3, 2). This means that f(−1) = f(1) = f(3) = 2. Three distinct inputs, −1, 1, and 3, produce the same output, 2. Therefore f is not one-to-one and does not have an inverse function.

10. Example: Determining if a function is one-to-one graphically (3 of 3)
Solution b. Because every horizontal line intersects the graph at most once, different inputs (x-values) always result in different outputs (y-values). Therefore f is one-to- one and has an inverse function.

11. Horizontal Line Test
If every horizontal line intersects the graph of a function f at most once, then f is a one-to-one function.

12. Increasing, Decreasing and One-to-One Functions
If a continuous function f is always increasing on its domain, then every horizontal line will intersect the graph of f at most once. By the horizontal line test, f is a one-to-one function. Similarly, if a continuous function g is only decreasing on its domain, then g is a one-to-one function.

13. Inverse Function
for every x in the domain of f and

14. Example: Finding and verifying an inverse function (1 of 5)
Let f be a one-to-one function given by f(x) = x³ − 2.

15. Example: Finding and verifying an inverse function (2 of 5)
Let f(x) = x³ − 2. Solution a. Solve y = f(x) for x.

16. Example: Finding and verifying an inverse function (3 of 5)
Let f(x) = x³ − 2.
Both the domain and the range of the cube root function include all real numbers. The graph of
is the graph of the cube root function shifted left 2 units.

17. Example: Finding and verifying an inverse function (4 of 5)

18. Example: Finding and verifying an inverse function (5 of 5)
and now

19. Finding a Symbolic Representation for f −1

20. Numerical Representation of Inverse (1 of 3)
In the top table, f computes the percentage of the U.S. population with 4 or more years of college in year x. x
1940
1970
2015
f(x) 5 11 34 x 5 11 34
f-1(x)
1940
1970
2015

21. Numerical Representation of Inverse (2 of 3)
Function f has an inverse function.

22. Numerical Representation of Inverse (3 of 3)
The domains and ranges are interchanged.
This relation does not represent the inverse function because input 4 produces two outputs, 1 and 2.

23. Domains and Ranges of Inverse Functions

24. Graphs of Functions and Their Inverses

25. Example: Representing an inverse function graphically (1 of 2)
Solution
Calculator display. [−5,5,1] by [−5,5,1]

26. Example: Representing an inverse function graphically (2 of 2)