Function notation Revision Notes for AQA GCSE Further Maths

Function notation

What is function notation?

Function notation is a mathematical way of representing relationships between inputs and outputs. When we write $f(x)$ , we're describing a function called "f" that takes an input value "x" and produces a specific output. Think of it as a mathematical machine that transforms one number into another following a set rule.

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The most important thing to remember is that a function must give exactly one output for every input. This means that for any value you put in, you'll always get the same result out.

Understanding function machines

A function can be visualised as a machine with a clear process. Let's look at how this works:

This flowchart shows a function machine that takes an input, squares it, then adds 2. This represents the function $f(x) = x^2 + 2$ .

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Worked Example: Function Machine in Action

Here's how it works with different inputs:

When we input 5: $5 \rightarrow 25 \rightarrow 27$ (so the output is 27)
When we input -2: $-2 \rightarrow 4 \rightarrow 6$ (so the output is 6)
When we input x: $x \rightarrow x^2 \rightarrow x^2 + 2$ (so the output is $x^2 + 2$ )

This process shows us that our function rule is $f(x) = x^2 + 2$ .

Writing and evaluating functions

Once we know our function rule, we can use function notation to find outputs for specific inputs. The key technique is substitution - wherever you see x in the function, replace it with your given value.

Using our example $f(x) = x^2 + 2$ :

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Worked Example: Evaluating f(5)

Step 1: Replace every x with 5 $f(5) = 5^2 + 2$

Step 2: Calculate following order of operations $f(5) = 25 + 2 = 27$

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Worked Example: Evaluating f(-2)

Step 1: Replace every x with -2 $f(-2) = (-2)^2 + 2$

Step 2: Calculate following order of operations $f(-2) = 4 + 2 = 6$

The uniqueness rule for functions

Not every mathematical relationship is a function. For something to be called a function, it must pass this crucial test: each input must produce exactly one output.

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Common Mistake to Avoid:

For example, $f(x) = ±x^2$ would not be a function because one input (like $x = 2$ ) would give two different outputs ( $±4$ ). This violates the fundamental rule that functions must have unique outputs.

Working with more complex functions

Let's examine how to handle different types of function problems with more complex expressions and operations.

Evaluating functions with different expressions

Consider $f(x) = 10 - 4x$ and $g(x) = x^3$ .

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Worked Example: Finding f(-1)

Step 1: Substitute -1 for x $f(-1) = 10 - 4(-1)$

Step 2: Simplify $f(-1) = 10 + 4 = 14$

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Worked Example: Finding g(1/2)

Step 1: Substitute 1/2 for x $g\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^3$

Step 2: Calculate the power $g\left(\frac{1}{2}\right) = \frac{1}{8}$

Finding expressions for composite inputs

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Worked Example: Finding f(3x) when f(x) = 10 - 4x

Step 1: Replace every x with 3x $f(3x) = 10 - 4(3x)$

Step 2: Simplify $f(3x) = 10 - 12x$

Solving function equations

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Worked Example: Solving g(x) = -64 when g(x) = x³

Step 1: Set up the equation $x^3 = -64$

Step 2: Solve for x using the cube root $x = \sqrt\[3]{-64} = -4$

Common exam techniques

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When working with function notation problems, remember these essential strategies that will help you tackle any function question confidently.

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Key Exam Techniques:

Always substitute carefully - replace every instance of the variable
Follow the order of operations - brackets, powers, multiplication/division, addition/subtraction
Check your uniqueness - remember that functions must give one output per input
Show your working clearly - substitute first, then calculate step by step

For solving function equations, isolate the variable using inverse operations, just like with regular algebraic equations.

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Remember!

Function notation $f(x)$ means "function f of x" - it shows what happens when you input x
Functions are like machines: one input always produces exactly one output
To evaluate $f(3)$ , substitute 3 for every x in the function rule
Not all mathematical relationships are functions - check the uniqueness rule
When solving function equations, treat them like regular algebraic equations once you've made the substitution

Function notation (AQA GCSE Further Maths): Revision Notes