Section 2.8Section 2.8

One-to-One Functions and Their Inverses

What is a Function?

Relationship between inputs (domaindomain) and outputs (rangerange) such that each input produces only one output

Passes the vertical line test

OK for outputs to be shared

One-to-One Functions

A function is a one-to-oneone-to-one function if no outputs are shared (each y-value corresponds to only one x-value)

Formal definition: If f(x1) = f(x2), then x1 = x2

Passes the horizontal line test

a bx

y

y = f (x)

(a, f (a)) (b, f (b))

x

y

(a, f (a))

y = f (x)

a

One-to-One Functions

Horizontal Line Test

A function is one-to-one if and only if each horizontal line intersects the graph of the function in at most one point.

x

y

x

y

x

y

(a) An increasing function is always one-to-one

(b) A decreasing function is always one-to-one

(c) A one-to-one function doesn’t have to be increasing or decreasing

Increasing and Decreasing Functions

{(1, 1), (2, 4), (3, 9), (4, 16), (2, 4)}

Yes, each y-value corresponds to a unique x-value.

{(-2, 4), (-1, 1), (0, 0), (1, 1)}

Example 1:

Are the following functions one-to-one?

No, there are 2 y-values of 1, which correspond to different x-values.

This is a modified slide from the Prentice Hall Website.

17

What is an Inverse Function?

FunctionMachine

5 5

FunctionMachine

g(5)

f(17)

( )g x

What is an Inverse Function? (cont’d)

FunctionMachine

x x

FunctionMachine

g( )

f [ ]g(x)

Inverse Function

An Inverse Function is a new function that maps y-values (outputs) to their corresponding x-values (inputs).

Notation for an inverse function is: f -1(x) For a function to have an inverse, it must pass the

horizontal-line test (i.e., only one-to-one functions have an inverse function!)

Why do we need a new function? Sometimes we have the y-value for a function and we want to know what x-value caused that y-value.

Formal definition of an Inverse Function

Let f be a function denoted by y = f(x). The inverse of f, denoted by f -1(x), is a function such that:

(f -1 ○ f )(x) = f -1[ f (x) ] = x for each x in the domain of f, and

(f ○ f -1 )(x) = f [ f -1(x) ] = x for each x in the domain of f -1.

3 5 5Verify that ( ) and ( ) are inverses.

3

xf x g x

x x

f g x fx

( ( ))

53

53 5

35

3

x

x

53 5

335 3

3

xxx

x

15 5 3

5x 5

5x

x

Example 2:

This is a modified slide from the Prentice Hall Website.

g f x gxx

( ( ))

3 5 53 5

3xx

53 5

3

xx xx

5

3 5 3x

x x 5

5x

x

This is a modified slide from the Prentice Hall Website.

Yes, these functions are inverses.

3 5 5Verify that ( ) and ( ) are inverses.

3

xf x g x

x x

Example 2 (cont’d):

How do we find an Inverse Function?

If a function f is one-to-one, its inverse can generally be found as follows:

1. Replace f(x) with y.

2. Swap x and y and then solve for y.

3. Replace y with f -1(x).

yx

5

3

xy

5

3

xy x 3 5xy x 3 5

yxx

3 5 f xxx

1 3 5( )

This is a slide from the Prentice Hall Website.

5Find the inverse of ( ) , 3

3f x x

x

Example 3:

Swap x and y.

Solve for y.

Replace f(x) with y

What does an Inverse Function look like?

Remember, to find an inverse function we just interchanged x and y.

Geometric interpretation: graph will be symmetrical about the line y = x.

Domain of f Range of f

Range of f 1 Domain of f 1

f 1

f

This is a slide from the Prentice Hall Website.

The domain of the original function becomes the range of the inverse function.

The range of the original function becomes the domain of the inverse function.