Finding Inverse of f(x) = 2x + 3

Pre-Calculus · Axiom Academy

EXAMPLE Finding the Inverse of f(x) = 2x + 3 Undo a linear function step by step using the swap-and-solve method. Find the inverse function f^ -1 (x) for f(x) = 2x + 3 . To do it, we rename f(x) as y , swap x and y , and solve the result for y . Nice work — you inverted a linear function by renaming, swapping, and solving. The method: set y = f(x) , swap x and y , then solve for y to get f^ -1 (x) . Inverses undo in reverse: f multiplies by 2 then adds 3 , so f^ -1 subtracts 3 then divides by 2 . The blue point on f at (0, 3) mirrors to the purple point on f^ -1 at (3, 0) — every point of f^ -1 is the reflection of a point of f across y = x . Swap-and-solve works for any one-to-one function — the reflection across y = x is what an inverse means geometrically.

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