Inverse Functions

Calculus Readiness · Axiom Academy

The undo button of functions: swap inputs and outputs, and watch the graph fold across the line y = x . 1. Undoing: the Inverse Runs the Machine Backward If a function f takes you from 3 to 11 , the inverse f^ -1 takes you back from 11 to 3 . Watch a value travel forward through f , then ride back through f^ -1 and land exactly where it started. 2. The Recipe: Swap x and y , Then Solve Finding an inverse from a formula is mechanical. Every point (a, b) on f becomes the point (b, a) on f^ -1 — so we literally swap x and y in the equation, then solve for y again. Watch one point make the swap. Write the function as y = f(x) . Solve the new equation for y . Worked example — a linear function Find the inverse of f(x) = 3x + 4 . Worked example — a rational function Find the inverse of . Write and swap: . Now clear the denominator and collect the y terms. 3. Not Every Function Has an Inverse Consider f(x) = x^2 . Both 3 and -3 map to 9 — so if you try to reverse it, what goes back from 9 ? It can't be both. An inverse must be a function (one output per input). The animation sweeps a horizontal line down the parabola: while it crosses twice , the function is not reversible. 4. The Graph of an Inverse: a Reflection Here is the beautiful fact. The graph of f^ -1 is the graph of f reflected across the line y = x . Reflecting across y = x swaps the x - and y -coordinate of every point — which is exactly what the inverse does. Watch f fold over the dashed mirror, point by point.

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