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Australia
10&10a

5.07 Inverses and their graphs

Worksheet
Inverse functions
1

Find the inverse of the following functions:

a

y = - 2 \left(x + 2\right)

b

y = \dfrac{4 x - 3}{5}

c

y = 6 - x^{3}

d

y = x^{5} + 2

e

y = \dfrac{1}{x - 8}

f

y = \dfrac{7 x + 6}{x - 7}

g

f \left(x\right) = 5 x

h

y = - 8 x + 6

i

y = \dfrac{2}{9} x

j

y = \dfrac{1}{8} x + 7

k

y = \dfrac{1}{7} x - 1

2

For each of the following pair of functions:

i

Evaluate f\left(g\left(x\right)\right).

ii

Evaluate g\left(f\left(x\right)\right).

iii

Are f \left( x \right) and g \left( x \right) inverses?

a

f \left( x \right) = - \dfrac{8}{9} x and g \left( x \right) = - \dfrac{9}{8} x

b

f \left( x \right) = 4 x + 16 and g \left( x \right) = \dfrac{1}{4} x - 4

c

f \left( x \right) = 3 x - 12 and g \left( x \right) = \dfrac{1}{3} x - 4

d

f \left( x \right) = x^{3} - 5 and g \left( x \right) = \sqrt[3]{x + 5}

Graphs of functions and inverses
3

State whether the following functions have an inverse function:

a
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
x
-3
-2
-1
1
2
3
4
5
y
b
-5
-4
-3
-2
-1
1
2
3
4
5
x
-3
-2
-1
1
2
3
4
5
y
c
-5
-4
-3
-2
-1
1
2
3
4
5
x
-3
-2
-1
1
2
3
4
5
y
d
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
e
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
f
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
g
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
4

In each graph below two functions and the line y = x are drawn. For each graph, state whether the functions are inverse functions of each other.

a
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
b
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
c
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
d
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
e
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
x
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
y
f
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
g
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
5

Sketch the graph of the inverse of the following functions:

a
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
b
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
c
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
d
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
e
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
f
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
6

Consider the graph of the function f \left( x \right) over the line y = x:

Sketch the graph of f^{ - 1 } \left(x\right).

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
7

Consider the graph of y = \dfrac{1}{x} over the line y = x:

a

Sketch the graph of the inverse of \\y = \dfrac{1}{x}.

b

Compare the inverse graph to the original graph.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
8

Consider the functions f \left( x \right) = \dfrac{1}{x} + 5 and g \left( x \right) = \dfrac{1}{x - 5}.

a

Sketch the graph of f \left( x \right).

b

Sketch the graph of g \left( x \right) on the same number plane.

c

Are f \left( x \right) and g \left( x \right) inverses?

9

Consider the graphs of f \left( x \right) and g \left( x \right):

a

State the equation of f \left( x \right).

b

State the equation of g \left( x \right).

c

Evaluate f \left( g \left( x \right) \right).

d

Evaluate g \left( f \left( x \right) \right).

e

State whether the following statements are correct:

i

g \left( x \right) is an inverse of f \left( x \right).

ii

f \left( g \left( x \right) \right) has gradient -\dfrac{2}{3}.

iii

f \left( x \right) is an inverse of g \left( x \right).

iv

g \left( f \left( x \right) \right) has gradient 1.

-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
Applications
10

A ski resort has introduced a new ski pass system whereby everyone needs to purchase an electronic card for a one time fee of \$8. They can then put credit on the card for each day's lift access. Lift access costs \$80 per day.

Let C \left(x\right) = 8 + 80 x represent the cost of purchasing x days worth of lift access on the card.

a

Define y, the inverse of C \left(x\right) = 8 + 80 x.

b

What does the inverse function y = C^{ - 1 } \left(x\right) represent?

c

Determine the number of days' lift access, y, that can be purchased with \$728.

11

The function t = \sqrt{\dfrac{d}{4.9}} can be used to find the number of seconds it takes for an object in Earth's atmosphere to fall d metres.

a

State the function for d in terms of t.

b

Find the distance a skydiver has fallen 5 seconds after jumping out of a plane.

12

The following formula can be used to convert Fahrenheit temperatures x to Celsius temperatures T \left( x \right):

T \left( x \right) = \dfrac{5}{9} \left(x - 32\right)

a

Find T \left( - 13 \right).

b

Find T \left( 86 \right).

c

Find T^{ - 1 } \left(x\right).

d

What can the formula T^{ - 1 } be used for?

13

The function d\left(t\right) = 120 - 4.9 t^{2} can be used to find the distance, d, that an object dropped from a height of 120 metres has fallen after t seconds, where t \geq 0.

a

Is the function d \left(t\right) one-to-one?

b

Find the inverse function, t \left(d\right).

c

Prove that d \left( t \left(d\right)\right) = d.

d

Prove that t \left( d \left(t\right)\right) = t.

e

Are d and t inverse functions?

f

How long it will take an object to fall 41.6 metres when dropped from a height of 120 metres?

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