Inverse Functions Summary

Pre-Calculus · Axiom Academy

Consolidating the unit: one-to-one functions, finding and verifying an inverse, reflecting across y = x , and restricting a domain to make an inverse exist. An inverse f^ -1 undoes f : if f sends , then f^ -1 sends . It exists only when f is one-to-one . A function is one-to-one exactly when it passes the horizontal line test — no horizontal line meets the graph more than once. The defining relationship is . The notation f^ -1 is not a reciprocal: . Find f^ -1 by swapping x and y and solving; verify by composing both directions; graph it by reflecting f across the line y = x . The domain and range trade places: and . Core Concept One-to-One & the Horizontal Line Test A function is one-to-one if different inputs always give different outputs — every output is hit by exactly one input. Only one-to-one functions have inverses, because the inverse needs a single unambiguous input to send each output back to. Test it: the graph passes the horizontal line test if no horizontal line crosses it more than once. Watch out for: f(x) = x^2 fails — f(-2) = f(2) = 4 — so it has no inverse without restricting the domain. Core Concept What an Inverse Is The inverse f^ -1 reverses f step for step: it undoes whatever f did. Composing a function with its inverse in either order returns the original input — that is the identity, and it is the whole definition. When to use: any time you need to solve for the input that produced a known output.

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