Function Composition

Learn how to compose functions and their properties.

Function Composition

If f: A → B and g: B → C, the composition g ∘ f is a function from A to C defined by (g ∘ f)(x) = g(f(x)).

g ∘ f means: first apply f, then apply g

Example: f(x) = 2x, g(x) = x + 3
  (g ∘ f)(x) = g(f(x)) = g(2x) = 2x + 3
  (f ∘ g)(x) = f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6

Composition Is Not Commutative

In general, g ∘ f ≠ f ∘ g. The order matters!

Example:
  f(x) = x², g(x) = x + 1
  (g ∘ f)(x) = x² + 1
  (f ∘ g)(x) = (x + 1)² = x² + 2x + 1

  g ∘ f ≠ f ∘ g

Composition Is Associative

(h ∘ g) ∘ f = h ∘ (g ∘ f)

This is always true. You can compose in any grouping:
  h(g(f(x))) = (h ∘ g ∘ f)(x)

Properties Preserved by Composition

If f and g are both injective, then g ∘ f is injective.
If f and g are both surjective, then g ∘ f is surjective.
If f and g are both bijective, then g ∘ f is bijective.

🧪 Quick Quiz

Given f(x) = 2x + 1 and g(x) = x², what is (f ∘ g)(3)?

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