How to Find a Composite Function

A composite function combines two functions into a single new function by using the output of one as the input of the other. In Add Math we write the composite of $f$ after $g$ as $fg(x)$, which means $f(g(x))$, the inner function $g$ acts first, then $f$ acts on that result.

The composite $fg$ is a genuinely new function with its own rule and its own graph, not a product of the two. This method answers questions such as "given $f$ and $g$, find $fg(x)$", "evaluate $gf(3)$", or "if $fg(x)$ is known, find the missing function".

It is a core Form 4 Functions skill, and it also underpins inverse functions, because a function composed with its inverse returns the input unchanged.

Understanding composition well is what lets you read the notation confidently. Two functions written side by side with no operation sign always mean composition, so $fg$ is never "$f$ times $g$".

Getting this straight early makes the whole Functions chapter far less confusing.

Reach for composition whenever a question gives you two functions and asks for a combined one written as $fg(x)$, $gf(x)$, $f^2(x)$, or asks you to evaluate a stacked expression like $fg(2)$. The tell-tale sign is two function definitions followed by a request that places one function inside another.

You also use composition when a question defines $fg(x)$ directly and asks you to find one of the original functions, or to solve an equation such as $fg(x)=k$. Because Functions opens the syllabus and composition appears in both Paper 1 and Paper 2, recognising the notation quickly saves time.

Whenever you see two function letters together with no plus, minus or multiplication sign between them, read it as composition and decide which function acts first.

Work in this order so the substitution never goes wrong:

1. 1

   Identify the inner and outer function

   In $fg(x)$, the inner function is $g$ (it acts first) and the outer function is $f$. Read the letters right to left.

2. 2

   Write out the outer function

   Write the outer rule with its input shown clearly, for example $f(\square)=2\square+1$, so you can see every slot the input goes into.

3. 3

   Substitute the whole inner function

   Replace every input slot of the outer function with the entire inner expression, keeping it inside brackets.

4. 4

   Expand and simplify

   Multiply out the brackets and collect like terms until you have one tidy expression in $x$.

5. 5

   Evaluate if a number is asked

   For a value like $fg(2)$, either substitute $x=2$ into your composite, or work inside-out: compute $g(2)$ first, then apply $f$.

Try this yourself first, then check each line against the solution.

The inside-out check in the note is worth building into a habit. Computing $g(2)$ first and then applying $f$ is a completely independent route to the same answer, so if the two methods disagree you know at once that a bracket or a sign has slipped somewhere.

• Reading $fg(x)$ as $f(x)\times g(x)$. It means $f(g(x))$, a substitution, never a product.
• Getting the order backwards. In $fg(x)$ the function $g$ acts first, so substitute $g$ into $f$, not the other way round.
• Substituting into only some of the $x$ terms. Every input slot of the outer function must be replaced by the whole inner expression.
• Dropping the brackets around the inner function, for example writing $2x^2-3+1$ instead of $2(x^2-3)+1$.
• Assuming $fg=gf$. They are usually different functions, so always compute the one the question actually asks for.

Most lost marks in composition come from a single decision, which function acts first, and from dropping the brackets in the very next line. In a one-to-one lesson our teacher watches you set up $fg(x)$ and stops the moment the wrong function goes inside, so the habit is corrected before it hardens.

Because our teachers are experienced, you get someone who explains why $g$ acts first, not just tells you to memorise it. Lessons are online and taught in English, while SPM papers are set in both Malay and English, so we make sure the notation reads the same to you in either version.

We also drill the inside-out check so a slip never survives to the final line.