Composite functions Revision Notes for AQA GCSE Further Maths

Composite functions

What are composite functions?

A composite function is created when you apply two or more functions one after another in sequence. Think of it like a mathematical assembly line where the output of one function becomes the input of the next function.

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The key thing to remember is that order matters when working with composite functions. Unlike addition or multiplication, changing the order of functions will usually give you a completely different result.

Understanding the notation

When you see $fg(x)$ , this means you apply function $g$ first, then apply function $f$ to the result. This might seem backwards at first, but think of it as reading from right to left - the function closest to the $x$ gets applied first.

Here's how it works step by step:

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Function Composition Process:

$fg(x) = f\[g(x)]$
First: apply $g$ to $x$ , giving you $g(x)$
Then: apply $f$ to that result, giving you $f\[g(x)]$

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Worked Example: Basic Function Composition

If $f(x) = x^2$ and $g(x) = x + 2$ , then:

$fg(x) = f\[g(x)] = f(x + 2) = (x + 2)^2$
Similarly, $gf(x) = g\[f(x)] = g(x^2) = x^2 + 2$

Notice how $fg(x)$ and $gf(x)$ give completely different results! This is why the order is so important.

Using flow charts to understand composition

A helpful way to visualise function composition is through flow charts. For the expression $(5x + 6)^4$ , you can break this down as:

$x \rightarrow 5x + 6 \rightarrow (5x + 6)^4$

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This flow chart shows that:

$g(x) = 5x + 6$ (applied first)
$f(x) = x^4$ (applied second)
So $fg(x) = (5x + 6)^4$

Combining functions with arithmetic operations

You can also combine functions using basic arithmetic operations:

Addition: $f(x) + g(x)$
Multiplication: $f(x) \times g(x)$
Division: $f(x) \div g(x)$

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Worked Example: Arithmetic Combinations

If $f(x) = x^2$ and $g(x) = x + 2$ :

$f(x) + g(x) = x^2 + (x + 2) = x^2 + x + 2$
$f(x) \times g(x) = x^2 \times (x + 2) = x^3 + 2x^2$
$f(x) \div g(x) = x^2 \div (x + 2) = \frac{x^2}{x + 2}$

Worked examples with detailed solutions

Let's look at a comprehensive example that shows different types of function composition.

This example demonstrates several important concepts:

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Worked Example: Multiple Function Compositions

For $fg(x)$ : Start with $g(x) = x^2$ , then apply $f$ to get $f(x^2) = 2x^2 - 3$

For $gf(x)$ : Start with $f(x) = 2x - 3$ , then apply $g$ to get $g(2x - 3) = (2x - 3)^2$

For $fgh(x)$ : This is a triple composition. First apply $h(x) = \frac{1}{x}$ , then $g$ to get $g\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^2 = \frac{1}{x^2}$ , then $f$ to get $f\left(\frac{1}{x^2}\right) = 2\left(\frac{1}{x^2}\right) - 3 = \frac{2}{x^2} - 3$

For $f^2(x)$ : This means $f(f(x))$ . Apply $f$ twice: $f(2x - 3) = 2(2x - 3) - 3 = 4x - 6 - 3 = 4x - 9$

Common exam tips and traps

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Critical Points to Avoid Mistakes:

Watch the order: The biggest trap is getting the order wrong. Always remember that in $fg(x)$ , you apply $g$ first, then $f$ .

Bracket carefully: When substituting one function into another, use brackets to avoid mistakes. For example, if $f(x) = x^2$ and you need $f(2x - 3)$ , write it as $(2x - 3)^2$ not $2x - 3^2$ .

Check your algebra: Function composition often involves expanding brackets or simplifying fractions. Take your time with the algebra to avoid careless errors.

Self-composition: When you see $f^2(x)$ , this means $f(f(x))$ , not $\[f(x)]^2$ . Apply the function to itself, don't square the result.

Remember!

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Key Points to Remember:

Composite functions involve applying functions in sequence - order is crucial
In $fg(x)$ , apply $g$ first, then $f$ (read from right to left)
Use brackets when substituting to avoid algebraic errors
Flow charts can help visualise the step-by-step process
$fg(x)$ and $gf(x)$ usually give different results - don't assume they're the same

Composite functions (AQA GCSE Further Maths): Revision Notes