Properties of Composition

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

Compositions can be regrouped without changing the order of evaluation, but generally cannot be reordered. Domain compatibility remains necessary at every stage.

Visual guide

VISUAL GUIDEA composition needs compatible outputs

x → x² → √(x²)

Squaring first maps −2 and 2 to 4; taking the square root next maps 4 to 2. Thus √(x²) = |x|, not x on all real inputs. The composite loses the sign information from the first map.

Method

(f\circ g)\circ h=f\circ(g\circ h).

Both sides mean $f(g(h(x)))$ . With the identity function $I(x)=x$ , $f\circ I=I\circ f=f$ on the appropriate domain.

Worked example

For $f(x)=x+1$ and $g(x)=x^2$ , $(f\circ g)(x)=x^2+1$ , while $(g\circ f)(x)=(x+1)^2$ . At $x=1$ , the outputs are $2$ and $4$ . Thus associativity does not imply commutativity. An inverse is special: composing it with the original function returns the identity on the corresponding set.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

For f(x)=x² and g(x)=x+1, do f∘g and g∘f always agree?

Yes: both functions have domain ℝ.

No: at x=1 they give 4 and 2.

Yes: composition is commutative.

Hint 1 · Find a starting point

Compare (x+1)² with x²+1.

Hint 2 · Take the next step

One counterexample is enough to disprove equality of functions.

Show the reasoning

Answer: No: at x=1 they give 4 and 2.

f(g(1))=4, whereas g(f(1))=2; order matters.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Common mistake

Canceling $f$ from $f(a)=f(b)$ requires injectivity. It is not a general algebraic cancellation rule.

Check your understanding

For $f(x)=2x$ , $g(x)=x-1$ , $h(x)=x^2$ , compute $f(g(h(x)))$ .

Show answer

$2(x^2-1)=2x^2-2$ .

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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