3.6 Inverse FunctionsObjectives:

1. Define inverse relations & functions.2. Find inverse relations from tables, graphs, & equations.3. Determine whether an inverse relation is a function.4. Verify inverses using composition.

Example #1Graphing an Inverse Relation

The graph of a function f is shown. Graph the inverse and describe the relationship between the function & its inverse.

x y

-6 5

-5 0

-4 -3

-3 -4

-2 -3

-1 0

0 5

Make a table of points from the figure.

Switch the x and y coordinates.

x y

5 -6

0 -5

-3 -4

-4 -3

-3 -2

0 -1

5 0

Graphs of Inverse Relations

x y

5 -6

0 -5

-3 -4

-4 -3

-3 -2

0 -1

5 0

Graph the new set of points.

The graph of the inverse f is a reflection of the graph of f across the line y = x.

Example #2Graphing an Inverse in Parametric Mode

Graph the function & its inverse in parametric mode.

f(x) 0.9x5 0.1x4 0.4x3 x 1

14.01.09.0)(

)(

:

345

ttttty

ttx

Original

tty

tttttx

Inverse

)(

14.01.09.0)(

:345

A. Find g(x), the inverse of f(x) 2x1

Example #3Finding an Inverse from an Equation

2

1

2

12

1

21

12

12

1

xxf

yx

yx

yx

xy This can be checked quickly by graphing the original and the inverse on the Y = screen.

This does not need parametric mode.

Example #3Finding an Inverse from an Equation

B. Find g(x), the inverse of 57 12)( xxf

12

12

12

12

75

57

57

57

yx

yx

yx

xy

7 51

7 5

75

12

12

12

xxf

yx

yx

To enter higher roots on the calculator, enter the root value first, then press MATH 5: x√

Example #4Finding the Inverse from an Equation

Find the inverse of f(x) 4x2 2x

4

411

8

4122

8

4142

8

1642

42

4422

24

240

24

24

1

2

2

2

2

xx

xxxf

xy

xcba

xyy

yyx

xxy

One-to-One Functions

A function is considered one-to-one if its inverse is also a function.

Use the horizontal line test to determine if the graph of the inverse will also be a function.

If the inverse is a function it is notated f -1. **This does not mean f to the -1 power.**

g(x) 0.5x3 x1

f(x) 3x5 2x4 3x3 x 2

h(x) 2 0.8x3

Example #5Using the Horizontal Line Test

Graph each function below and determine whether it is one-to-one.

A.

B.

C.

Yes

Yes

No

Example #6Restricting the Domain

Find an interval on which the function is one-to-one, and find f -1 on that interval.

f(x) x2 3

2 4 6 8 10–2–4–6–8–10 x

2

4

6

8

10

–2

–4

–6

–8

–10

y

The function is one-to-one from [0, ∞). Using this domain the inverse would be the positive square root of x. Alternatively, if (−∞, 0] is chosen, negative square root of x is the inverse.

xxf

xy

yx

yx

xy

1

2

2

2

3

3

3

2 4 6 8 10–2–4–6–8–10 x

2

4

6

8

10

–2

–4

–6

–8

–10

y

Composition of Inverse Functions

A one-to-one function and its inverse have these properties.

Also, any two functions having both properties are one-to-one and inverses of each other.

For every x in the domain of f and f -1

xxff

xxff

1

1

Example #7Verifying the Inverse of a Function

Verify that f and g are inverses of each other.

f(x) 2x3 x

g(x) 3xx 2

xx

xx

x

x

x

x

xx

x

x

x

x

x

x

xx

xx

xgf

6

1

1

6363

2

2

6

2

323

2

6

2

33

2

6

23

3

23

2

xx

xx

x

x

x

x

xx

x

x

x

x

x

x

xx

xx

xfg

6

1

1

6262

3

3

6

3

322

3

6

23

2

3

6

23232

3

Since the both compositions of the functions equal x, then the functions are inverses of each other.