1. Finding Inverse Functions
Section 4.3
Finding Inverse Functions

2. Objectives:
1. To find the inverse of a one-to-
one function algebraically and
graphically.
2. To determine whether two
functions are inverses by
applying composition.

3. The inverse of a function, symbolized f-1, is a relation that results when the first and second coordinates of each ordered pair are interchanged.
f = {(3, 5), (2, 9), (-6, 4), (5, 7)}
f-1 = {(5, 3), (9, 2), (4, -6), (7, 5)}

4. Finding the Inverse of a Function
1. Interchange the variables (use y instead of f(x)).
2. Solve for y and if the inverse is a function, express it using f-1(x).

5. EXAMPLE 1 Find the inverse function rule for f(x) = 5x – 7.
y = 5x – 7
x = 5y – 7
x + 7 = 5y
x + 7 5
y =
f-1(x) = x + 1 5 7

6. f
f-1
y = x

7. EXAMPLE 2 Find f ◦ f-1 if f(x) = 5x – 7.
f ◦ f-1 (x) = f(f-1(x))
x + 7 5
= f( )
x + 7 5
= 5( ) - 7
= x + 7 – 7
= x

8. EXAMPLE 3 Graph the inverse of the function shown.

9. Homework pp

10. ►A. Exercises f-1 = {(7, 4), (-3, 2), (7, 5), (8, 1)}
Give the inverse relation of f = {(4, 7),
(2, -3), (5, 7), (1, 8)}. State whether the inverse is a function.
1. f = {(4, 7), (2, -3), (5, 7), (1, 8)}
f-1 = {(7, 4), (-3, 2), (7, 5), (8, 1)}
f-1 is not a function

11. ►A. Exercises
Determine whether the inverse relation is a function. If it is, write it in function rule notation. If the inverse relation is not a function, show why the original function is not one-to-one.
3. f(x) = -4x + 6

12. ►A. Exercises
Determine whether the inverse relation is a function. If it is, write it in function rule notation. If the inverse relation is not a function, show why the original function is not one-to-one.
7. y = x2 – 8x + 16

13. ►B. Exercises
Determine whether the rules given are truly inverse functions.
11. f(x) = 3x + 9
f-1(x) =
x – 9 3

14. ►B. Exercises
Determine whether the rules given are truly inverse functions.
13. h(x) =
h-1(x) =
x + 2 7

15. ►B. Exercises
Find f-1(x) for each function, and draw the graph of both f and f-1 on the same Cartesian plane.
19. f(x) = , x  -2 1
x + 2

16. ►B. Exercises 19. f(x) = , x  -2 1 x + 2 f(x) = 1 x + 2
y + 2
= y + 2 1 x
y = – 2 1 x

17. 19.

18. ►B. Exercises Graph the function and its inverse on the same graph.
21. f(x) = -3x – 4

19. ►B. Exercises
Graph the inverse of each function.
25.

20. ■ Cumulative Review
Use the function f(x) = -2 sin (3x – ) to answer the following questions.
33. What is the period?  2 2 3
p = =
|n|

21. ■ Cumulative Review
Use the function f(x) = -2 sin (3x – ) to answer the following questions.
34. What is the amplitude?  2
A = |A| = |-2| = 2

22. ■ Cumulative Review
Use the function f(x) = -2 sin (3x – ) to answer the following questions.
35. What is the phase shift?  2
/2 3
P. S. = = = b n  6

23. ■ Cumulative Review
Use the function f(x) = -2 sin (3x – ) to answer the following questions.
36. What are the zeros?  2 k 3 +  6
for k  Z

24. ■ Cumulative Review
Use the function f(x) = -2 sin (3x – ) to answer the following questions.
37. Graph f(x).  2 - 