An inverse undoes a one-to-one function on its range. If f(x)=y, then f-1(y)=x. The domain and range exchange, and (a,b) reflects to (b,a) in y=x. f-1(f(x))=x on D_f, while f(f-1(x))=x on R_f. The notation f-1(x) does not mean 1f(x). Example y=3x-4 x= y+43 f-1(x)= x+43 Both domains and ranges are R. In general, do not replace f(x)=f-1(x) with f(x)=x without justification: for f(x)=-x, the two graphs coincide away from y=x too.

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 \(\dfrac{1}{\mathrm{f}(x)}\).

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

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.

Functions

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Inverse functions

Inverse functions