2.3PureStretch

Composite Functions

A composite function applies one function and then another: $f g (x)$ means “do $g$ first, then $f$ ”. Order matters, and the inner function’s output must be valid input for the outer one.

25 min Video by Zeeshan Zamurred Functions and Graphs

Edexcel A level Maths: 2.3 Composite FunctionsWatch the full walkthrough before the notes below.

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What you'll be able to do

Understand composite function notation
Evaluate fg(x) and gf(x)
Recognise that order matters
Build composite functions algebraically

Inside out

The composite $f g (x)$ means $f (g (x))$ — apply the $inner$ function $g$ first, then feed its output into $f$ . Work from the inside out.

f g (x) = f (g (x))

g

first, then

f

g (2) = 4

f (4) = 5

Answer

5

Order matters

In general $f g (x) \neq = g f (x)$ — swapping the order changes the result. Always check which function is written first (it acts last).

f (2) = 3

g (3) = 9

Answer

9

(≠

f g (2) = 5

)

Tip — fg means f outside, g inside. The function nearest the x acts first.

Building the expression

To get $f g (x)$ as a formula, substitute the whole of $g (x)$ into $f$ wherever $x$ appears, then simplify.

1Replace

x

f

with

g (x) = x^{2}

2 (x^{2}) + 1

Answer

2 x^{2} + 1

Formula recap

f g (x) = f (g (x))

Inner function first.

f g (x) \neq = g f (x) in general

Order matters.

f f (x) = f (f (x))

A function composed with itself.

Common mistakes to avoid

Doing f first in fg(x).

fg(x) = f(g(x)) — the inner function g acts first.

Assuming fg(x) = gf(x).

They are usually different; check the order.

Key takeaways

fg(x) = f(g(x)): apply the inner function first.
Order matters — fg and gf are generally different.
Build the formula by substituting g(x) into f.

Test yourself

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