Composition

Function composition: (f ∘ g)(x) = f(g(x)) means apply g first, then f. Example: f(x) = 2x+1, g(x) = x². Then (f ∘ g)(x) = f(x²) = 2(x²)+1 = 2x²+1. Also (g ∘ f)(x) = g(2x+1) = (2x+1)² = 4x²+4x+1. Note: f ∘ g ≠ g ∘ f (order matters). Inverse composition: (f ∘ f⁻¹)(x) = x = (f⁻¹ ∘ f)(x). Finding f⁻¹: (1) Set y = f(x); (2) Solve for x; (3) Swap x and y. Example: f(x) = 2x+3. y = 2x+3 → x = (y-3)/2 → f⁻¹(x) = (x-3)/2. Verify: f(f⁻¹(x)) = f((x-3)/2) = 2((x-3)/2)+3 = x. Exam tip: Domain of g must include range of f for f ∘ g. Practice: Complex compositions.