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iGCSE (2021 Edition)

4.01 Function notation

Worksheet
Function notation
1

The point \left(7, - 6 \right) satisfies the function f(x). Write this using function notation.

2

If f \left(x\right) = 9 x^{2} + 7 x - 4, find:

a

f(-4)

b

f(10)

3

If f \left(x\right) = - 6 x + 4, find:

a

f \left( 4 \right)

b

f \left( 0 \right)

4

If f(x) = 4 x + 4, find:

a

f \left( 2 \right)

b

f \left( - 5 \right)

5

If f \left( x \right) = 3 x - 1, find:

a

f \left( 3 \right)

b

f \left( - 4 \right)

6

If g \left( x \right) = \dfrac{7 x}{4}, find:

a

g \left( 5 \right)

b

g \left( - 4 \right)

7

Consider the function f \left( x \right) = 4 + x^{3}.

a

Evaluate f \left( 4 \right)

b

Evaluate f \left( - 2 \right)

8

Consider the function f \left( x \right) = 2 x^{2} - 2 x + 5. Evaluate f\left(\dfrac{1}{2}\right).

9

Consider the function f \left( x \right) = \sqrt{ 5 x + 9}. Find the exact value of:

a

f \left( 0 \right)

b

f \left( 2 \right)

c

f \left( - 1 \right)

10

If f \left( t \right) = \dfrac{t^{3} + 27}{t^{2} + 9} , evaluate the following:

a

f \left( - 3 \right)

b

f \left( 3 \right)

c

f \left( 4 \right)

11

Consider the function f \left( x \right) = x^{2} + 8 x. Write an expression for:

a

f \left( a \right)

b

f \left( b \right)

12

Consider the function f \left( x \right) = 2 x^{3} + 3 x^{2} - 4.

a

Evaluate f \left( 0 \right).

b

Evaluate f \left( \dfrac{1}{4} \right).

13

If j(x) = 3^{x} - 3^{ - x }, find the following, rounding your answers to two decimal places if necessary:

a

j(0)

b

j(1)

c

j(4)

14

If m \left( x \right) = \sqrt{12^{2} - x^{2}}, find:

a

m \left( 4 \right)

b

m \left( 0 \right)

c

m \left( 9 \right)

d

m \left( \sqrt{2} \right)

15

Consider the function p \left( x \right) = x^{2} + 8.

a

Evaluate p \left( 2 \right).

b

Form an expression for p \left( m \right).

16

Consider the equation x + 3 y = 6.

a
Make y the subject of the equation.
b

Rewrite the equation using function notation f \left( x \right) for y.

c

Find the value of f \left( 3 \right).

17

Consider the equation x - 4 y = 8.

a
Make y the subject of the equation.
b

Rewrite the equation using function notation f \left( x \right).

c

Find the value of f \left( 12 \right).

18

Consider the equation y + 6 x^{2} = 3 - x.

a
Make y the subject of the equation.
b

Rewrite the equation using function notation f \left( x \right).

c

Find the value of f \left( 3 \right).

19

Consider the equation y - 4 x^{2} = 5 + x.

a
Make y the subject of the equation.
b

Rewrite the equation using function notation f \left( x \right).

c

Find the value of f \left( 2 \right).

20

Consider the equation - 6 x + 5 y = 7.

a
Make y the subject of the equation.
b

Rewrite the equation using function notation f \left( x \right).

c

Find the value of f \left( 3 \right).

21

Use the graph of the function f \left( x \right) to find each of the following values:

a

f \left( 0 \right)

b

f \left( - 2 \right)

c

The value of x such that f \left( x \right) = 3

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
22

Consider the relation h \left(x\right) = - x^{2} + 6 x - 6.

a

For any input value of x, state the maximum number of distinct output values h \left(x\right) can produce.

b

Is h \left(x\right) a function? Explain your answer.

23

For each of the following pairs of variables, determine which is the independent variable and which is the dependent variable:

a

Cost of pizza and size of pizza.

b

Mark achieved on a test and time spent studying.

c

Duration of a loan and amound borrowed.

d

Time spent exercising and physical fitness level.

24

Consider the function f(x) = x^{2} - 49.

a

Find:

i

f(1)

ii

f(8)

iii

f(0)

b

If f(x) = 12, what are the possible values for x, rounded to two decimal places?

25

Consider the function g \left( x \right) = a x^{3} - 3 x + 5.

a

Form an expression for g \left( k \right).

b

Form an expression for g \left( - k \right).

c

Is g \left( k \right) = g \left( - k \right)?

d

Is g \left( k \right) = - g \left( - k \right)?

Applications
26

If Z(y) = y^{2} + 12 y + 32, find y when Z(y) = - 3.

27

A function f \left( x \right) is defined by f \left( x \right) = \left(x + 4\right) \left(x^{2} - 4\right).

a

Evaluate f \left( 6 \right).

b

Find all solutions for which f \left( x \right) = 0.

28

A graph of a quadratic equation of the form y = a x^{2} + b x + c passes through the points \left(0, - 7 \right), \left( - 1 , - 8 \right) and \left(4, 77\right) .

a

Using the point \left(0, - 7 \right), find the value of c.

b

Substitute c and \left( - 1 , - 8 \right) into the equation y = a x^{2} + b x + c to obtain an equation that describes a in terms of b.

c

Substitute c and \left(4, 77\right) into the equation y = a x^{2} + b x + c to obtain a simplified second equation in terms of a and b.

d

Solve for a and b.

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Outcomes

0606C1.1

Understand the terms: function, domain, range (image set), one-one function, inverse function and composition of functions.

0606C1.2A

Use function notation. e.g. f(x) = sin x, f: x ↦ sin x, f(x) = lg x or f: x ↦ lg x.

0606C6.1

Solve simple simultaneous equations in two unknowns by elimination or substitution.

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