Restricting a Domain to Invert

Inverting a function means swapping inputs and outputs: f⁻¹ undoes whatever f did. That swap only works cleanly if f never sends two different inputs to the same output. y = x² breaks this immediately, because x = 3 and x = -3 both give y = 9. If you tried to invert that, the inverse would have to send 9 back to both 3 and -3, and a function can't output two values for one input. So the full parabola has no inverse.

The fix is to cut the domain down to a piece where each output really does come from only one input. For y = x², restricting to x ≥ 0 keeps the right half of the parabola, where every y-value traces back to exactly one x-value. Now inversion works: solve for x and keep the root that matches the restricted domain. For f(x) = x² - 4 with x ≥ 0, write y = x² - 4, add 4 to get x² = y + 4, then take the square root. Since x ≥ 0, you keep only the positive root: x = √(y + 4). Swapping variable names gives f⁻¹(x) = √(x + 4).

This is exactly why square root and logarithm functions exist as the tame, single-valued inverses of squaring and exponentials: someone already restricted the domain for you. It also matters once a function models something real, because the math restriction (like x ≥ 0) might not be the only limit. A square's side length can't be negative, but it also can't exceed whatever material or space the context allows, so the inverse's domain can shrink further than the algebra alone demands.