If \(f(x) = x + 3\), find \(f(2)\).
Substitute x = 2 into the function.
\(f(2) = 2 + 3 = 5\).
Functions
Functions describe how one value depends on another using input-output relationships. Function notation is an important foundation for advanced algebra.
Overview
A function is a rule that takes an input and gives exactly one output.
You can think of it as a machine: something goes in, the rule happens, and something comes out.
\( f(x) = 2x + 3 \)
This means the function f takes a number, doubles it, and adds 3.
What you should understand after this topic
- Understand what a function is
- Understand how input and output work
- Interpret what f(x) means
- Substitute values into a function rule
- Find missing inputs or outputs
- Understand how function machines link to algebra
Key Definitions
Function
A rule that gives exactly one output for each input.
Input
The value that goes into the function.
Output
The value that comes out after applying the rule.
Function Notation
Symbols such as \( f(x) \) used to show the function rule and output.
Rule
The operation or formula used inside the function.
Inverse Function
A rule that reverses what the original function does.
Key Rules
Read \( f(x) \)
It means “the output of function \( f \) when the input is \( x \)”.
Substitute carefully
Replace every \( x \) in the rule with the given number.
Use brackets
If substituting a negative number, always use brackets.
One input, one output
A function gives only one output for each input value.
How to Solve
What is a function?
A function takes an input, applies a rule, and gives an output. You can think of it like a function machine.
Input \( \rightarrow \) rule \( \rightarrow \) output
\( 4 \rightarrow 2(4)+1 = 9 \)
If the rule is “multiply by 2 and add 1”, then input 4 gives output 9.
Key idea: The same input should always give the same output.
Reading function notation
Function notation tells you which value to substitute into the function.
\( f(x) = 3x - 4 \)
\( f(2) = 3(2) - 4 = 2 \)
\(f(2)\) means substitute \(x = 2\).
This uses the same skill as substitution.
Substitute carefully
When substituting negative values, use brackets to avoid sign errors.
\( f(x) = x^2 + 3 \)
\( f(-2) = (-2)^2 + 3 = 4 + 3 = 7 \)
Important
Write negative inputs in brackets. Without brackets, squaring can be misread.
Find an output from a function
To find an output, substitute the input value and calculate.
\( f(x) = 5x + 2 \)
\( f(6) = 5(6) + 2 = 32 \)
Exam tip: Show the substitution step clearly before simplifying.
Find a missing input
Sometimes you know the output and need to work backwards by solving an equation.
\( f(x) = 2x + 7 \), and \( f(x) = 15 \)
\( 2x + 7 = 15 \)
\( x = 4 \)
This links to solving linear equations.
Understand inverse functions
An inverse function undoes the original function by reversing the operations.
Function: multiply by 3, then add 2
Inverse: subtract 2, then divide by 3
\( f(x) = 3x + 2 \)
\( f^{-1}(x) = \dfrac{x - 2}{3} \)
Exam tip: Reverse the operations in the opposite order.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on function notation, substitution, and inverse functions.
Edexcel
Given \( f(x) = 2x + 3 \), find \( f(4) \).
Edexcel
Given \( g(x) = x^2 - 5 \), find \( g(-2) \).
Edexcel
If \( f(x) = 3x - 1 \), find \( f(a) \).
Edexcel
Given \( f(x) = x + 4 \) and \( g(x) = 2x \), find \( fg(3) \).
Edexcel
Find the inverse function of \( f(x) = 3x - 7 \).
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on composite functions, inverse functions, and algebraic reasoning.
AQA
Given \( f(x) = 2x + 1 \) and \( g(x) = x^2 \), find \( f(g(3)) \).
AQA
Given \( f(x) = x - 4 \) and \( g(x) = 3x \), find \( gf(x) \).
AQA
Find the inverse of \( f(x) = 5x + 2 \).
AQA
The function \( f(x) = x^2 \) is defined for \( x \geq 0 \). Explain why this restriction is necessary when finding the inverse.
AQA
Given \( f(x) = 2x - 3 \), find the value of \( x \) for which \( f(x) = 11 \).
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising function machines, interpretation, and algebraic manipulation.
OCR
A function machine multiplies a number by 4 and then adds 3.
OCR
Write the function in the form \( f(x) \).
OCR
Find \( f(5) \).
OCR
The function \( f(x) = 3x - 2 \). Find the value of \( x \) when \( f(x) = 13 \).
OCR
Given \( f(x) = x + 1 \) and \( g(x) = 2x - 3 \), find \( fg(x) \).
OCR
Explain the difference between \( fg(x) \) and \( gf(x) \).
Exam Checklist
Step 1
Read the function rule carefully.
Step 2
Substitute the input into every \( x \).
Step 3
Use brackets for negative numbers.
Step 4
Simplify in the correct order.
Most common exam mistakes
Notation
Thinking \( f(x) \) means multiplication.
Negative inputs
Forgetting brackets around a negative number.
Inverse order
Undoing operations in the wrong order.
Missing input
Not forming an equation when the output is given.
Common Mistakes
These are common mistakes students make when working with functions in GCSE Maths.
Not substituting into every x
Incorrect
A student replaces only one occurrence of x in the function.
Correct
When substituting a value into a function, replace every instance of x. For example, in \(f(x) = 2x + x^2\), both terms must be updated.
Not using brackets for negative inputs
Incorrect
A student writes \(f(-3) = -3^2\) instead of \((-3)^2\).
Correct
Always use brackets when substituting negative numbers. This ensures operations like squaring are applied correctly.
Misinterpreting function notation
Incorrect
A student thinks \(f(x)\) means \(f \times x\).
Correct
\(f(x)\) represents the value of the function f when the input is x. It is not multiplication.
Getting the order wrong in inverse functions
Incorrect
A student reverses operations in the wrong order.
Correct
To find an inverse function, reverse the original steps in the correct order. Work backwards carefully to avoid mistakes.
Squaring negative numbers incorrectly
Incorrect
A student calculates \(-3^2 = 9\) instead of \(-9\).
Correct
Without brackets, the square applies only to the 3, so \(-3^2 = -9\). To square a negative number fully, write \((-3)^2 = 9\).
Try It Yourself
Practise evaluating and interpreting mathematical functions.
Foundation
Foundation Practice
Understand function notation and evaluate simple functions.
If \(f(x) = x + 7\), find \(f(3)\).
Replace x with 3.
\(f(3) = 3 + 7 = 10\).
If \(f(x) = 2x\), find \(f(5)\).
Multiply 2 by x.
\(f(5) = 2 × 5 = 10\).
If \(f(x) = 3x\), find \(f(4)\).
Multiply 3 by 4.
\(f(4) = 3 × 4 = 12\).
If \(f(x) = x^2\), find \(f(3)\).
Square the number.
\(f(3) = 3^2 = 9\).
If \(f(x) = x^2\), find \(f(-2)\).
Square -2.
\(f(-2) = (-2)^2 = 4\).
Which is correct if \(f(x) = x + 1\)?
Substitute x = 3.
\(f(3) = 3 + 1 = 4\).
If \(f(x) = x - 2\), find \(f(10)\).
Subtract 2 from 10.
\(f(10) = 10 - 2 = 8\).
A student says \(f(3) = f × 3\). What is wrong?
Function notation is not multiplication.
f(3) means replace x with 3, not multiply f by 3.
If \(f(x) = x + 5\), find \(f(0)\).
Substitute x = 0.
\(f(0) = 0 + 5 = 5\).
Higher
Higher Practice
Work with function rules, inverse thinking and composite functions.
If \(f(x) = 2x + 3\), find \(f(4)\).
Multiply first, then add.
\(f(4) = 2(4) + 3 = 8 + 3 = 11\).
If \(f(x) = 3x + 1\), find \(f(5)\).
Substitute x = 5.
\(f(5) = 3(5) + 1 = 16\).
If \(f(x) = x^2 + 1\), find \(f(2)\).
Square then add 1.
\(f(2) = 2^2 + 1 = 5\).
If \(f(x) = x^2 + 2x\), find \(f(3)\).
Substitute into both terms.
\(f(3) = 9 + 6 = 15\).
If \(f(x) = x + 2\) and \(g(x) = 3x\), find \(f(g(2))\).
Find g(2) first, then apply f.
\(g(2) = 6\), then \(f(6) = 6 + 2 = 8\).
If \(f(x) = x + 4\) and \(g(x) = 2x\), find \(f(g(3))\).
Work inside out.
\(g(3) = 6\), then \(f(6) = 10\).
If \(f(x) = 2x\), find \(f^{-1}(8)\).
Reverse the operation.
If \(f(x) = 2x\), then \(f^{-1}(x) = x/2\). So \(f^{-1}(8) = 4\).
If \(f(x) = x + 3\), find \(f^{-1}(10)\).
Undo +3.
Inverse is \(f^{-1}(x) = x - 3\). So \(10 - 3 = 7\).
Which is the inverse of \(f(x) = x + 5\)?
Reverse the operation.
To reverse +5, subtract 5.
If \(f(x) = 2x - 1\), find \(f^{-1}(7)\).
Undo -1, then divide by 2.
Add 1: 8, then divide by 2 → 4.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is a function?
A rule that maps inputs to outputs.
What does f(x) mean?
It represents the output when x is the input.
How do I evaluate a function?
Substitute the value into the function expression.