Properties of Composition — Cheat sheet

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Key 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.

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.

Avoid this mistake

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