Question

If h(x) has an inverse function h–1(x), then (h ∘ h^–1)(x) = (h^–1∘ h)(x).

True

False

Answer 1

False.

Reason: h∘h⁻¹ is the identity on the range of h, while h⁻¹∘h is the identity on the domain of h. These need not be the same set.

Counterexample: h(x) = e^x (domain R, range (0,∞)). Then
- (h∘h⁻¹)(x) = x for x > 0 (domain (0,∞)),
- (h⁻¹∘h)(x) = x for all real x (domain R).
They are not the same function (their domains differ).

(They would be equal if h were a bijection from a set onto itself, so both compositions equal the identity on the same set.)