Composition of functions is commutative: f∘g = g∘f for all f, g.

Solution

Maths sir · 3 steps

One counterexample kills a 'for all' claim, so try f(x) = x² and g(x) = x + 1.

1. Read the order. (f ° g)(x) = f(g(x)) means apply g first, then f. In (g ° f)(x) = g(f(x)) the order is reversed, so there is no reason for the two to match.
2. Pick a pair. Take f(x) = x² and g(x) = x + 1. Then (f ° g)(x) = (x+1)² and (g ° f)(x) = x² + 1.

   (f ° g)(1) = (1+1)² = 4 ≠ 2 = 1² + 1 = (g ° f)(1)

3. So. The two composites give different outputs at x = 1, so f ° g ≠ g ° f in general. The statement is false.

The idea to keep. Composition depends on order: f ° g means do g first. It is associative but not commutative, and one counterexample is enough to disprove a 'for all' statement.

Answer from the NTA final answer key. Solution by PrepWiser.