Finding the Inverse of a Function — Cheat sheet

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Key method

Write $y=f(x)$ , solve for $x$ in terms of $y$ , then rename the input of the inverse. Domain and range exchange roles. Verify by composition on the appropriate domains.

Example

For $f(x)=3x-5$ , solve $y=3x-5$ to get $x=(y+5)/3$ . Hence $f^{-1}(x)=(x+5)/3$ . For $g(x)=x^2$ on $x\ge0$ , take the nonnegative branch: $g^{-1}(x)=\sqrt x$ , also defined for $x\ge0$ .

Avoid this mistake

Writing both $\pm\sqrt x$ does not define a single inverse function; the original domain decides which branch to use.