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1.5 Inverse Functions

Example 1: An equation for determining the cost of a taxi ride is: C = 0.75n + 2.50, where n is the number of kilometers and C is the cost.

n C10

20

30

40

$10.00

$17.50

$25.00

$32.50

Complete the table of values.

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The inverse function allows us to ask the question: if I have $50, how far can I travel?

original function: C = 0.75n + 2.50

× 0.75 + 2.50n C

input number of kilometers

output Cost

inverse function

÷ 0.75 – 2.50n C

output number of kilometers

input Cost2.50

0.75

Cn

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A Bf

6 23

24

715

Domain: {2, 4, 6}

Range: {7, 15, 23}

We know that a function maps elements of a domain onto elements of a range.

A B

6 23

24

715

f –1

The inverse function maps the elements of the range back onto the elements of the domain.

Domain: {7, 15, 23}

Range: {2, 4, 6}

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Assume we have a function which consists of a set of ordered pairs.

f(x) = {(1, 5), (2, 9), (3, 13), (4, 17)}

Domain: {1, 2, 3, 4} Range: {5, 9, 13, 17}

f –1(x) = {(5, 1), (9, 2), (13, 3), (17, 4)}

Range: {1, 2, 3, 4}Domain: {5, 9, 13, 17}

f –1(x) means the inverse function of f(x).

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Suppose we have the following relation f(x) consisting of the following points. Determine the graph of f –1(x).

We have symmetry about the line y = x.

f(– 4) =

f(0) =

?

?

2

5

f –1(1) = ?

f –1(0) = ?

–2

–6

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Example 2: Given the graph of y = f(x) below, sketch the graph of y = f –1(x).

Step 1: Sketch the graph of y = x.

Step 2: Map the points using the line y = x as the axis of symmetry.

Step 3: Join the points

D: x and R: y 0

f(2) = ?8

f –1(2) = ?0

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f –1(x)

D: x 0 and R: y

f(x)

D: x and R: y 0

State the domain and range of f –1(x)

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f(x) = 3x + 4Example 3: Given the equation:

Determine the equation of f – 1(x).

y = 3x + 4f f –1

1. Replace y by x and x by y.

2. Isolate y.

x – 4 = 3y

4

3

xy

x = 3y + 4

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Compare f and f –1 and the order in which operations are carried out.

1. Multiply by 3

2. Add 4

1. Subtract 4

2. Divide by 3

y = 3x + 4

f f –1

4

3

xy

You will notice the order and the operations are inverted.

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When you go to bed at night

1- you untie your laces

2- you take off your shoes 2- you put on your shoes

When you get up in the morning

1- you put on your socks

Notice the inverse operation

3- you take off your socks 3- you tie up your laces

Reverse order … reverse operation

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1. Multiply by 5

2. Subtract 2

1. Add 2

2. Divide by 5

f(x) = 5x – 2 f –1

1 2( )

5

xf x

Example 4: Determine the inverse of f(x) = 5x – 2

x + 2

2

5

x

Ex: f(4) = 5(4) – 2= 20 – 2= 18

f –1(4) =4 2

5

6

5

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Example 5: Given:3

( ) ( 4)2

f x x

a) Determine f –1(x)

3( 4)

2x y

2 3( 4)x y

2 3 12x y

2 12

3

xy

1 2 12

( )3

xf x

replace x by y and y by x.

× 2

isolate y.

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1 2 12( )

3

xf x

3

( ) ( 4)2

f x x

1 2( 1) 12( 1)

3

bf b

2 2 12

3

b

2 14

3

b

b) Determine f (–6)

3( 6) ( 6 4)

2f

3( 10)

2

15

c) Determine f –1(b + 1)

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Example 6: the relation f is defined by 2x – 3y = 6.

Graph f

x f(x)

Using the intercept method.

0 – 203

Graph f –1

f -1(x)x

0– 20 3

f

f –1

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The relation f is defined by 2x – 3y = 6.

f

f –1

Determine:

f(–3) = – 4

f –1(–3) = – 1.5

f –1(x):

2y – 3x = 6

2y = 3x + 6

33

2y x

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Inverse of a Quadratic Function

• A) Find the inverse of f(x)= x2-1

• B) Graph f(x) and its inverse.

• C) Is the inverse of f(x) a function?

• D) Determine the domain and range of f(x) and its inverse, i.e. f-1(x)