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Use Formal Function Notation

In this worksheets, students will learn to use formal function notation.

'Use Formal Function Notation' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Notation, Vocabulary and Manipulation, Language of Functions

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

A function is a rule that connects inputs and outputs. Every input has one corresponding value of an output.

f(x) means a function of x. x is the input

f(x) = x2 + 3 means a functions of x where you square x and then add 3 to it.

If you want to work out the output when the input is 3, we write it like this f(3)= 32 + 3 = 9 + 3 = 12.

 

Sometimes the input is an algebraic term. If this is the case, replace each letter x with the desired input.

Let's write the function written above f(x) = x2 + 3 as a function with input 4x:

f(4x) = (4x)2 + 3 = 16x2 + 3

f(x) = 2x + 4

Match correct inputs with outputs.

Column A

Column B

f(1)
6
f(0)
18
f(-1)
4
f(7)
2

f(2) = 5

Which of the functions will five you this answer?

f(x) = x² + 1

f(x) = 4x - 3

f(x) = x² - 1

f(x) = 3x - 4

g(x) = x2 + 5x

Find the value of g(3).

21

62

24

59

h(x) = x2 - 5x + 1

Find the value of h(2).

-47

-5

-7

15

Here are three functions:

f(x) = 6x - 2

g(x) = x² - 3x

h(x) = x² + 2x - 1

Match the functions to the correct outputs.

Column A

Column B

f(0)
7
g(-1)
4
h(2)
-1
f(-3)
-2
g(3)
-20
h(-2)
0

f(x) = 3x - 7

Find the value of x for which f(x) = 26.

x=99

x=57

x= 6.3

x=11

g(x) = 3x + 1

Find the value of x for which g(x) = 19

x = 6

x = 19

x = 54

x = 17

If h(x) = x² - 3x what is h(5x)?

 

5x² - 15x

5x² - 3x

25x² - 3x

25x² - 15x

If f(x) = x² - 1, match the correct expressions for each of the new functions.

Column A

Column B

f(2x)
36x² - 1
f(6x)
x&sup4; - 1
f(x+1)
4x² - 1
f(x²)
x² + 2x

f(x) = x² + 1

Find f(x&sup2, + 1).

x&sup4; + 2

x&sup4; + 2x² + 1

x&sup4; + 2x² + 2

x&sup4; + 1

  • Question 1

f(x) = 2x + 4

Match correct inputs with outputs.

CORRECT ANSWER

Column A

Column B

f(1)
6
f(0)
4
f(-1)
2
f(7)
18
EDDIE SAYS
To find which output corresponds to each input, replace x with the number that is in the bracket next to f. f(1) = 2 × 1 + 4 = 2+4=6 f(0) = 2 × 0 + 4=0+4=4 f(-1) = 2 × -1 + 4=-2+4=2 f(7) = 2 × 7 + 4=14+4=18
  • Question 2

f(2) = 5

Which of the functions will five you this answer?

CORRECT ANSWER
f(x) = x² + 1
f(x) = 4x - 3
EDDIE SAYS
Replace x with 2 and see if the output is 5. This only works for f(x) = x² +1 and f(x) = 4x - 3.
  • Question 3

g(x) = x2 + 5x

Find the value of g(3).

CORRECT ANSWER
24
EDDIE SAYS
Here the function has a different name (g), but we still do the same to work out the output. Replace x with 3. g(3) = 3² + 5 × 3 = 9 + 15 = 24
  • Question 4

h(x) = x2 - 5x + 1

Find the value of h(2).

CORRECT ANSWER
-5
EDDIE SAYS
Remember to replace x with 2. h(2) = 2² - 5 × 2 + 1 = 4 - 10 + 1 = -5
  • Question 5

Here are three functions:

f(x) = 6x - 2

g(x) = x² - 3x

h(x) = x² + 2x - 1

Match the functions to the correct outputs.

CORRECT ANSWER

Column A

Column B

f(0)
-2
g(-1)
4
h(2)
7
f(-3)
-20
g(3)
0
h(-2)
-1
EDDIE SAYS
Double-check what to replace x with in each function. Let's have a look at a couple of trickier ones. h(-2)= (-2)² + 2 × (-2) - 1 = 4 - 4 - 1 = -1 g(3) = 3² - 3×3 = 9 - 9 = 0
  • Question 6

f(x) = 3x - 7

Find the value of x for which f(x) = 26.

CORRECT ANSWER
x=11
EDDIE SAYS
Here you need to work backwards. 3x - 7 = 26 3x = 26 + 7 3x = 33 x = 33/3 x = 11
  • Question 7

g(x) = 3x + 1

Find the value of x for which g(x) = 19

CORRECT ANSWER
x = 6
EDDIE SAYS
You've got the value of the output, so work backwards using inverse operations to find the input. 3x + 1= 19 3x = 19 - 1 3x = 18 x = 18/3 x = 6 So the correct input is 6.
  • Question 8

If h(x) = x² - 3x what is h(5x)?

 

CORRECT ANSWER
25x² - 15x
EDDIE SAYS
25x² - 15x is the correct answer. You need to replace each x in h(x) with a 5x. This gives you: (5x)² - 3(5x) = 25x² - 15x.
  • Question 9

If f(x) = x² - 1, match the correct expressions for each of the new functions.

CORRECT ANSWER

Column A

Column B

f(2x)
4x² - 1
f(6x)
36x² - 1
f(x+1)
x² + 2x
f(x²)
x&sup4; - 1
EDDIE SAYS
To find the expression for f(x+1), replace each x with (x + 1). (x+1)² - 1 Now expand and simplify: x² + 2x + 1 - 1 = x² Similarily for f(x²), replace each x with x² (x²)² - 1 = x&sup4; - 1
  • Question 10

f(x) = x² + 1

Find f(x&sup2, + 1).

CORRECT ANSWER
x&sup4; + 2x² + 2
EDDIE SAYS
This is a tricky question! You need to change the input into x² + 1. f(x², + 1) = (x², + 1)² + 1 Expand the brackets and simplify: x&sup4; + x² + x² + 1 + 1 = x&sup4; + 2x² + 2
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