Restricting Domains

Pre-Calculus · Axiom Academy

Why a function like y = x^2 has no inverse — and how chopping its domain to a one-to-one piece brings the inverse back. 1. The Problem with f(x) = x^2 A function has an inverse only if it's one-to-one — every output comes from exactly one input. The quick visual test is the horizontal line test : if any horizontal line meets the graph more than once, the function fails. Watch the line y = 4 drop onto the parabola. It strikes the curve at two points, x = -2 and x = 2 , because (-2)^2 = 2^2 = 4 . Two inputs, one output. If we tried to define f^ -1 (4) , should it be 2 or -2 ? An inverse must return a single value, so f(x)=x^2 over all real numbers has no inverse function . We don't have to give up. Keep only the right half of the parabola — restrict the domain to . The animation chops away the part with and keeps the bold piece on the right. Now send the horizontal line down again: against the kept piece it meets the curve exactly once at every height. The restricted function passes the horizontal line test, so it's one-to-one — and one-to-one functions have inverses. Domain: all real numbers. Fails the line test — no inverse. Domain: . Passes the line test — invertible. To find the inverse of the restricted piece, set y = x^2 and solve for x . Since , we take the non-negative root: . Renaming gives the inverse function. the restricted function (kept piece)

This is the written version of the interactive lesson above. See the full Pre-Calculus course.