Function Composition: Feeding One Function Into Another

Composition is what happens when two rules run back to back. Convert pounds to euros, then apply a booking fee. Take a measurement, then square it. The output of the first becomes the input of the second, and that chain is a function in its own right.

The Idea

The ring symbol is read “composed with”, but says the same thing and is usually clearer.

The critical point is order: the inner function runs first.

Written , the sits on the left but acts second. That is exactly the same convention as arithmetic brackets: in you do the bracket first. Reading left to right is the instinct to fight here.

Evaluating at a Number

Work from the inside out, one step at a time.

With and , find :

1. Inner first: .
2. Outer second: .

So .

Doing it in two written steps is worth the extra line. Trying to do both at once is where sign and bracket errors creep in.

Building the Composed Formula

To get a general expression, substitute the whole of into wherever appears.

The brackets are essential. squares its entire input, so it is , not .

A useful habit: write first, then drop into the box. It makes the substitution mechanical.

Order Matters

Composition is not commutative. Swapping the order almost always gives a different function.

With the same and :

Check at : the first gives 9, the second gives 5. Genuinely different.

There is one important case where the order does not matter: when is the inverse of . Then both compositions return the input unchanged, which is precisely what “inverse” means.

The Domain of a Composition

Two conditions must hold at once, and the second is the one people miss.

1. must be in the domain of the inner function .
2. must be in the domain of the outer function .

Worked case. and , for :

• needs .
• needs its input , so , which rules out .

Domain: .

Looking only at the composed formula would catch the exclusion, but a formula that simplifies can hide the inner restriction entirely — which is why checking separately is the safer routine. See Domain.

Decomposing a Function

The reverse skill: given a complicated function, find two simpler ones that compose to make it. This is essential for the chain rule in calculus.

Look for an expression being acted on as a whole. That inner expression is ; whatever is done to it is .

Decompositions are not unique — could also be with . The useful one is normally the most obvious inner bracket.

Composition vs Multiplication

These two get confused constantly, so it is worth being explicit.

At : composition gives 9, multiplication gives 12. See Combining Functions for the arithmetic operations.

Worked Example A: Numbers Both Ways

, . Find and .

• : , then .
• : , then .

Different, as expected.

Worked Example B: The Composed Formula

, . Find and .

Worked Example C: A Negative Inside a Square

, . Find .

, then .

The bracket around matters — without it the square would come out negative.

Worked Example D: Domain of a Composition

, . Find and its domain.

accepts everything, but needs a non-negative input, so . Domain: .

Worked Example E: Decomposition

Write as a composition.

The bracket is cubed as a whole, so and . Check: . ✓

Worked Example F: A Real Chain

A shop takes 20% off, then adds 5% tax on the reduced price.

A £50 item costs £42. Reversing the order, — here the two happen to agree, because both are simple multiplications. That is the exception, not the rule.

Common Mistakes to Avoid

• Running the outer function first. In , goes first.
• Assuming . They differ in almost every case.
• Confusing composition with multiplication. .
• Dropping brackets on substitution. with is .
• Checking only the composed formula for the domain. The inner function’s restriction still applies.
• Reading as multiplication. The ring is composition, not a product.
• Forgetting brackets around a negative intermediate value.

Where Composition Leads Next

• Inverse functions, defined by — the one case where order does not matter.
• The chain rule in calculus, which differentiates a composition and depends on decomposing correctly.
• Transformations, since is a composition with a shift.
• Unit conversion chains, where each step is a function applied to the last result.
• Programming, where piping the output of one function into another is the same idea.

Practice Problems

Use and for Problems 1–4.

Problem 1. Find .

Problem 2. Find .

Problem 3. Find .

Problem 4. Find .

Problem 5. For and , find .

Problem 6. For the same functions, find .

Problem 7. For and , find the domain of .

Problem 8. Decompose .

Problem 9. Decompose .

Problem 10. For and , find and .

Problem 11. A price rises 10%, then a £5 voucher is deducted. Write the final price as a composition of and .

Problem 12. Using the same functions, is the same?

Quick Reference

Composition is one of two ways to build a function from two others — the arithmetic route is Combining Functions. It underpins inverse functions and relies on function notation and domain. More Algebra lessons are available.