1. Chapter 5: Inverse, Exponential, and Logarithmic Functions
5.1 Inverse Functions 5.2 Exponential Functions 5.3 Logarithms and Their Properties 5.4 Logarithmic Functions 5.5 Exponential and Logarithmic Equations and Inequalities 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions

2. One-to-One Functions
Is the function a one-to-one function? If so, create the inverse function 𝑓 −1 𝑥 = Numerically (Table):
Age (years)
Height (inches) 5 40 9 45 13 60 17 66 19

3. One-to-One Functions
Is the function a one-to-one function? If so, create the inverse function 𝑓 −1 𝑥 = Numerically (Table):
Age (years)
Height (inches) 5 40 9 45 13 60 17 66 19
Height (inches)
Age (years) 40 5 45 9 60 13 66 17 19

4. One-to-One Functions
Given a word description, what would the inverse be:
You go into class, sit down, get your book out of your backpack, and open your book.

5. One-to-One Functions
Given a word description, what would the inverse be:
You go into class, sit down, get your book out of your backpack, and open your book.
Inverse:
You close your book, put your book in your backpack, get up and exit the classroom.

6. Horizontal Line Test
If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one. Example Use the horizontal line test to determine whether the graphs are graphs of one-to-one functions.

7. Horizontal Line Test
If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one. Example Use the horizontal line test to determine whether the graphs are graphs of one-to-one functions. No
Yes

8. Graphing the inverse relation or function
Draw the inverse relation, then the inverse function. No
Yes

9. One-to-One Functions

10. Inverse Functions
Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if

11. Inverse Functions
Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if

12. Finding an Equation for the Inverse Function Verbally and Algebraically
Finding the Equation of the Inverse of y = f(x)
For a one-to-one function f defined by an equation y = f(x), find the defining equation of the inverse as follows. (Any restrictions on x and y should be considered.)

13. Example of Finding f -1(x)

14. Example of Finding f -1(x)

15. Example of Finding f -1(x)
1. h(x) = 5 𝑥 + 9
2. f(x) = 4x3 - 5

16. Finding the Inverse of a Function with a Restricted Domain

17. Finding the Inverse of a Function with a Restricted Domain
Solution Notice that the domain of f is restricted to [–5, ), and its range is [0, ). It is one-to-one and thus has an inverse.

18. Application of Inverse Functions
Example Use the one-to-one function f(x) = 3x + 1 and the numerical values in the table to code the message PLAY MY MUSIC. A 1 B 2 C 3 D 4 E 5 F 6 G 7 H 8 I 9 J 10 K 11 L 12 M 13 N 14 O 15 P 16 Q 17 R 18 S 19 T 20 U 21 V 22 W 23 X 24 Y 25 Z 26

19. Application of Inverse Functions
Example Use the one-to-one function f(x) = 3x + 1 and the numerical values in the table to code the message PLAY MY MUSIC. A 1 B 2 C 3 D 4 E 5 F 6 G 7 H 8 I 9 J 10 K 11 L 12 M 13 N 14 O 15 P 16 Q 17 R 18 S 19 T 20 U 21 V 22 W 23 X 24 Y 25 Z 26