Inverse functions

An inverse undoes a one-to-one function on its range. If \(\mathrm{f}(x)=y\), then \(\mathrm{f}^{-1}(y)=x\). The domain and range exchange, and \((a,b)\) reflects to \((b,a)\) in \(y=x\).

\(\mathrm{f}^{-1}(\mathrm{f}(x))=x\) on \(D_\mathrm{f}\), while \(\mathrm{f}(\mathrm{f}^{-1}(x))=x\) on \(R_\mathrm{f}\). The notation \(\mathrm{f}^{-1}(x)\) does not mean \(\dfrac{1}{\mathrm{f}(x)}\).

Example

\(\begin{aligned}y&=3x-4\\x&=\dfrac{y+4}{3}\\f^{-1}(x)&=\dfrac{x+4}{3}\end{aligned}\)

Both domains and ranges are \(\mathbb R\). In general, do not replace \(\mathrm{f}(x)=\mathrm{f}^{-1}(x)\) with \(\mathrm{f}(x)=x\) without justification: for \(\mathrm{f}(x)=-x\), the two graphs coincide away from \(y=x\) too.