Graphs of Inverse Functions

Pre-Calculus · Axiom Academy

LESSON Graphs of Inverse Functions Why the graph of an inverse is just the original graph, folded across the line y = x . 1. The Graph Folds Across y = x Take a simple one-to-one function, f(x) = 2x + 1 , and the line y = x (the dashed crease). The graph of f^ -1 is what you get by reflecting the graph of f across that line — fold the page along y = x and the blue line lands exactly on the purple one. Its inverse f^ -1 (solve y = 2x+1 for x ) 2. Every Point Trades Its Coordinates Reflecting across y = x does one thing to a point: it swaps the two coordinates . Watch (1, 3) on f glide across the mirror and land on (3, 1) — and that landing point is on f^ -1 . It works because f(1) = 3 means f^ -1 (3) = 1 . (-1, -1) reflects onto itself — it already sits on the mirror y = x . Any point where f(x) = x is its own reflection, so it lies on f and f^ -1 at once. For f(x) = 2x+1 , solving 2x+1 = x gives x = -1 , the fixed point (-1,-1) . 3. The Two Graphs Meet on the Mirror Because f^ -1 is the reflection of f , any place the two graphs cross has to sit on the line y = x — a point off the mirror and its reflection are two different points. So to find where f meets f^ -1 , just solve f(x) = x . Set them equal: for f(x) = 2x+1 , the meeting point is found from 2x + 1 = x , which gives x = -1 , i.e. the point (-1, -1) . A point on f that isn't on y = x reflects to a different point on f^ -1 — so the graphs are apart there.

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