A function assigns each input of its domain exactly one output. An inverse exists only if the function is one-to-one on its domain — restrict x² to x ≥ 0 first. Composition fg(x) = f(g(x)) applies g first, and fg and gf generally differ. Graph transformations read from the factored form a·f(b(x − h)) + k: scale by 1/b and a, then shift by h and k.
For f(x) = x² − 4 with domain x ≥ 0, find the inverse and its domain using the same sequence: known relation → rearrange → select branch → state domain and range. (Answer: from y = x² − 4, x = √(y + 4) using the x ≥ 0 branch, so f⁻¹(x) = √(x + 4) with domain x ≥ −4 and range y ≥ 0.)
The function f(x) = x² has an inverse over all real numbers. This claim is false: two inputs share each output (2 and −2 both give 4), so the reverse mapping is not a function — the domain must first be restricted, for example to x ≥ 0, giving f⁻¹(x) = √x.
