1.7 Composite Functions

2026 📋 Syllabus Objectives

By the end of these notes, you should be able to:

  • Form composite functions — combine two functions into one new function
  • Use composite functions — evaluate and simplify composite functions
  • Understand that order matters — recognise that fg(x) and gf(x) are usually not the same thing

📖 Main Content

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What is a Composite Function?

In maths, a function is like a machine — you put a number in, it does something to it, and gives you a number out.

A composite function is what happens when you put two function-machines together in a row — the output of the first machine becomes the input of the second machine.

Simple analogy: Imagine a car wash. The first machine soaps the car, the second machine rinses it. The full car wash is a "composite" of both machines — one after the other.

The Notation: What Does fg(x) Mean?

When you see fg(x), read it like this:

"Apply g first, then apply f to the result."

In other words:

fg(x)=f(g(x))fg(x) = f\big(g(x)\big)

The function on the right (g) always goes first. The function on the left (f) always goes second.

This trips up many students — so remember: right to left, just like reading Arabic or Hebrew!

A Mapping Diagram to Visualise It

Think of it as a three-step journey:

x  ──[g]──▶  g(x)  ──[f]──▶  fg(x)
  • Start with x
  • Apply g → you get g(x)
  • Apply f to that result → you get fg(x)

The big "shortcut" arrow that goes all the way from x to fg(x) in one step is the composite function fg.

When Does a Composite Function Exist?

Not every pair of functions can be combined. For fg to exist (i.e. to be possible):

The range of g must fit inside the domain of f.

  • The range of a function = all the output values it can produce.
  • The domain of a function = all the input values it is allowed to receive.

So the output of g must be acceptable input for f. If g produces values that f cannot handle, then fg cannot be formed.

⚠️ Key Rule: Order Matters!

This is one of the most important ideas in this topic:

fg(x) is usually NOT the same as gf(x)

  • fg(x): apply g first, then f
  • gf(x): apply f first, then g

These are two different composite functions. Swapping the order almost always gives a different answer.

Think of it this way: Getting dressed in the morning — putting on your socks and then shoes is very different from putting on your shoes and then socks!

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