1.8 Inverse Functions

Any function can be represented by a set of ordered pairs.

For example: f(x) = x + 5→ goes from the set A = {1, 2, 3, 4} to the set B

{5, 6, 7, 8}

This can also be represented by:f(x) = x + 5: {(1, 5), (2, 6), (3, 7), (4, 8)}

Inverse Functions

• An inverse function, denoted by , is found by interchanging the first and second coordinates of each ordered pair.

f(x) = x + 4: {(1, 5), (2, 6), (3, 7), (4, 8)}

= : {(5, 1), (6, 2), (7, 3), (8, 4)}

1f

)(f 1 x

x f(x)

Domain of f(x) Range of f(x)

-1f of Range -1f ofDomain

If functions are inverses:

• If f(x) and g(x) are two functions that are inverses of each other, then:

a) (f ○ g) (x) = x

b) (g ○ f) (x) = x

• To verify that these 2 functions are inverses, you must show that f(g(x)) = x and g(f(x)) = x

f(x) = x + 4 g(x) = x - 4

f(g(x))= f(x - 4)= x - 4 + 4= x

g(f(x))= g(x + 4)= x + 4 - 4= x

Verify that the functions are inverses:

a) f(x) = 3x – 2 g(x) = x + 2

b) f(x) = x – 3 g(x) = 12 + 4x

c) f(x) = 2x + 4 g(x) = x - 2

31

41

21

Finding inverses informally:

ofeffect thehas (x),ffunction inverse The -1

f(x).function theundoing""

2 x f(x) offunction inverse theFind :ex What does this function do?How can this be “undone”?

Find the inverses of the following:

a) f(x) = x – 3

b) f(x) = 7x

c) f(x) =

d) f(x) =

4x

53-2x

Graphs of Inverse Functions:

• If the point (a, b) lies on the graph of f(x), then the point (b, a) must lie on the graph of the inverse function.

• This means that the graph of is a reflection of the graph of f(x) over the line

y = x

)(f 1 x

• Graph the function and its inverse.

f(x) = x + 2

)(f 1 x

• Graph the function and its inverse.

f(x) = 2x - 3

)(f 1 x

Do all functions have an inverse?

Think about what an inverse function actually does

Horizontal Line Test

• A function f has an inverse if and only if no horizontal line intersects the graph of f at more than one point.

Does the function have an inverse?

Does the function have an inverse?

Does the function have an inverse?

One-to-One Function

• A function is said to be one-to-one if for every input, there is exactly one output and for every output, there is exactly one input.

• If a function is one-to-one, the function has an inverse

f(x) = x²x f(x) Does every input have

exactly 1 output?

Does every output have exactly 1 input?

The function is not 1-1

-2 4-1 10 01 123

49

f(x) = x³x f(x)-2 -8-1 -10 01 12 83 27

Does every input have exactly 1 output?

Does every output have exactly 1 input?

The function is 1-1

1.8 Inverse Functions

Finding Inverse Functions Algebraically

Find the inverse of the function:

235)( xxf

For complicated functions, it is best to find the inverse function algebraically.

Finding Inverse Functions Algebraically

1) Use the horizontal line test to determine whether f has an inverse

2) In the equation f(x), replace f(x) with y3) Interchange the roles of x and y, then solve

for y4) Replace y with in the new equation5) Verify your answer

)(f 1 x

235)( xxf

1) Does it pass the horizontal line test?

)(xf

2) Replace f(x) with y

235 x

y

3) Interchange x and y and solve for y

x235 y

2) Replace f(x) with y

325 x

y

3) Interchange x and y and solve for y4) Replace y with 1f

1f

5) Verify your answer

Find the inverse of the following functions.

75xf(x)

2x4-xf(x)

3 1f(x) x

75xf(x)

3 1f(x) x

2x4-xf(x)

Review 1.7 & 1.8

• Basic operations on functions• Composition of functions• Domain of functions (interval notation)• Finding Inverses• Verifying Inverses• Graphing functions vs. inverses• Domains of inverses

One-to-One Function

• A function is said to be one-to-one if for every input, there is exactly one output and for every output, there is exactly one input.

• If a function is one-to-one, the function has an inverse

f(x) = x³x f(x)-2 -8-1 -10 01 12 83 27

Does every input have exactly 1 output?

Does every output have exactly 1 input?

The function is 1-1

f(x) = x²x f(x) Does every input have

exactly 1 output?

Does every output have exactly 1 input?

The function is not 1-1

-2 4-1 10 01 123

49

Interval Notation

• Identify the domain of the function using interval notation:

14)( xxf

Graph the function and determine whether or not it has an inverse

14)( xxf

Find the inverse of the function

14)( xxf

Verify the inverse of the function

14)( xxf