Inverse of function f exists if and only if f is:

f⁻¹ exists iff f is bijection. We have (f⁻¹ ∘ f)(x) = x. So (f⁻¹ ∘ f)(x) = (f ∘ f⁻¹)(y). e.g., f(x) = x³ is bijective; f⁻¹(y) = y^(1/3). f(x) = x² is not.