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Use Formal Function Notation

In this worksheets, students will learn to use formal notation to express and calculate inputs and outputs of functions.

'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 as an output.

 

f(x) means a function of x, where x is a specific 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

 

 

 

In this activity, we will use formal notation (shown in the examples above) to express and calculate inputs and outputs of functions. 

Consider the function below:

 

f(x) = 2x + 4

 

Based on this, match the correct inputs with outputs in the list below.

Column A

Column B

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

Consider the function below:

 

f(2) = 5

 

Which of the options in the list below will give you this answer?

f(x) = x2 + 1

f(x) = 4x - 3

f(x) = x2 - 1

f(x) = 3x - 4

Consider the function below:

 

g(x) = x2 + 5x

 

Based on this, find the value of g(3).

Consider the function below:

 

h(x) = x2 - 5x + 1

 

Based on this, find the value of h(2).

Here are three functions:

 

f(x) = 6x - 2

g(x) = x² - 3x

h(x) = x² + 2x - 1

 

Match each function below to its correct output.

Column A

Column B

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

Consider the function below:

 

f(x) = 3x - 7

 

Based on this, find the value of x for which f(x) = 26.

Consider the function below:

 

g(x) = 3x + 1

 

Based on this, find the value of for which g(x) = 19.

If h(x) = x² - 3x, what is h(5x)?

 

5x2 - 15x

5x2 - 3x

25x2 - 3x

25x2 - 15x

Consider the function below:

 

f(x) = x² - 1

 

Based on this, match the correct inputs with their functions in the list below.

Column A

Column B

f(2x)
36x2 - 1
f(6x)
x4 - 1
f(x + 1)
x2 + 2x
f(x²)
4x2 - 1

Consider the function below:

 

f(x) = x² + 1

 

Based on this, find f(x2 + 1).

x4 + 2

x4 + 2x2 + 1

x4 + 2x2 + 2

x4 + 1

  • Question 1

Consider the function below:

 

f(x) = 2x + 4

 

Based on this, match the correct inputs with outputs in the list below.

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, we need to replace x with the number that is in the bracket next to f in each case. 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 Did you find those matches successfully?
  • Question 2

Consider the function below:

 

f(2) = 5

 

Which of the options in the list below will give you this answer?

CORRECT ANSWER
f(x) = x2 + 1
f(x) = 4x - 3
EDDIE SAYS
f(2) = 5 means we need to find the functions above which give an output of 5 when x is replaced with 2. Work through each one, substituting 2 for x in each case: f(2) = x2 + 1 = 22 + 1 = 5 which matches f(2) = 5. f(x) = 4x - 3 = 4 × 2 - 3 = 5 which also matches. f(x) = x2 - 1 = 22 - 1 = 3 which does not match. f(x) = 3x - 4 = 3 × 2 - 4 = 2 which is also not a match. Did you find those two options which work?
  • Question 3

Consider the function below:

 

g(x) = x2 + 5x

 

Based on this, find the value of g(3).

CORRECT ANSWER
24
EDDIE SAYS
Here the function has a different name (g), but we still need to follow exactly the same process to work out the output. Let's replace x with 3 in the function: g(3) = 32 + 5 × 3 = 9 + 15 = 24
  • Question 4

Consider the function below:

 

h(x) = x2 - 5x + 1

 

Based on this, find the value of h(2).

CORRECT ANSWER
-5
- 5
EDDIE SAYS
Let's do the same again and 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 each function below to its correct output.

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 we need to replace x with in each function. It is also important to consider which function you need to work on e.g. f(0) means substitute 0 for x in the first function in the list. 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 Can you work out the remaining options independently to find their matching outputs?
  • Question 6

Consider the function below:

 

f(x) = 3x - 7

 

Based on this, find the value of x for which f(x) = 26.

CORRECT ANSWER
11
EDDIE SAYS
This question is asking us to find the value for the input (x) which gives an output of 26, so we need to work backwards. So 3x - 7 = 26 3x = 26 + 7 3x = 33 x = 33 ÷ 3 x = 11 So the correct input to reach an output of 26 is 11.
  • Question 7

Consider the function below:

 

g(x) = 3x + 1

 

Based on this, find the value of for which g(x) = 19.

CORRECT ANSWER
6
EDDIE SAYS
We've got the value of the output again, so we need to 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 to reach an output of 19 is 6.
  • Question 8

If h(x) = x² - 3x, what is h(5x)?

 

CORRECT ANSWER
25x2 - 15x
EDDIE SAYS
This may seem more challenging, as we need to work with algebra not numbers, but we just need to follow the same process. To find the correct answer, we need to replace each x in h(x) with a 5x. This gives us: h(5x) = (5x)² - 3(5x) = 25x² - 15x
  • Question 9

Consider the function below:

 

f(x) = x² - 1

 

Based on this, match the correct inputs with their functions in the list below.

CORRECT ANSWER

Column A

Column B

f(2x)
4x2 - 1
f(6x)
36x2 - 1
f(x + 1)
x2 + 2x
f(x²)
x4 - 1
EDDIE SAYS
To find the expression for f(x + 1), replace each x in the function with (x + 1): (x + 1)2 - 1 Now expand and simplify: x2 + 2x + 1 - 1 = x2 + 2x Similarly for f(x2), replace each x with x2: (x2)2 - 1 = x4 - 1 Can you find the other two matches independently, using these examples to support you?
  • Question 10

Consider the function below:

 

f(x) = x² + 1

 

Based on this, find f(x2 + 1).

CORRECT ANSWER
x4 + 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: x4 + x2 + x2 + 1 + 1 = x4 + 2x2 + 2 Congratulations on completing this challenging activity - there were some tricky questions to tackle here, so feel proud of yourself!
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