Section 9.1 – Inverse Functions

DOES an inverse function exist?

IF YES, you can find the inverse function.

The Existence of the Inverse of f(x)

IF for every x there is at most one y (function)

AND

IF for every y there is at most one x (one-to-one)

then an inverse function of f(x) exists.

The inverse function is denoted by 1f x

Graphical Existence of InversePasses BOTH vertical and horizontal line test.

Inverse Exists No Inverse Exists(1, 1), (-1, 1)

No Inverse Exists(1, 0), (0, 0)

Inverse Exists No Inverse Exists(3, 0.9), (7, 0.9)

Inverse Exists Inverse Exists

No Inverse Exists(-3, 2), (0, 2)

Inverse Exists

JanuaryFebruaryMarchJuly

WinterSpringSummer

t 0 1 2 3f(t) 5 2 2 4

Does an inverse exist?

FordBushCarterClinton

PresidentVice-President

2, 4 , 3, 2 , 1, 5

6, 5 , 8, 1 , 6, 3

No Inverse Exists(Jan, Winter), (Feb, Winter)

No Inverse Exists(1, 2), (2, 2)

Inverse Exists

No Inverse Exists(Ford, President),

(Ford, Vice-President)No Inverse Exists

(6, 5), (6, 3)

Algebraic Existence of Inverse

2y x 6x 4 2x y x y 7

2x y 4 2 2x y 16

No Inverse Exists(-4, -12), (-2, -12)

No Inverse Exists(4, 2), (4, -2)

Inverse Exists

No Inverse Exists(0, 2), (0, -2)

No Inverse Exists(4, 0), (-4, 0)

FINDING the inverse which exists

2. Switch the x and the y.

3. (Algebraically) Solve for y.

4. Replace y with 1f x

1. Determine if inverse function exists. If yes, proceed to #2.

Finding the inverse function GRAPHICALLY

Finding the Inverse Function TABULARLY

x -2 3 5 7 8 9f(x) 0 -1 4 9 -3 2

x 0 -1 4 9 -3 2-2 3 5 7 8 9 1f x

x 2 3 4 5 6 7f(x) -7 -1 3 4 8 4

x 2 3 4 5 6 7f(x) -7 -1 3 4 8 4

No Inverse Exists(5, 4), (7, 4)

Finding the Inverse ANALYTICALLY

x y 7

y 7x y 7 x

1f 7x x

f x 2x 3 x 2 3y

x y32

1 xx 32

f

f x x 1

x 1y 2x 1y

2x y1

12 xx 1 f

3f x x 3 3x 3y 3x y3

3 1 xx 3 f

3f x x 4 3x 4y

3x y4

3x y4

3 1 x4x f