Finding Inverse Functions Section 4.3 Finding Inverse Functions

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.

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)}

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).

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

f f-1 y = x

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

EXAMPLE 3 Graph the inverse of the function shown.

Homework pp. 185-187

►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

►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

►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

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

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

►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

►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

19.

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

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

■ 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|

■ 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

■ 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

■ 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

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