Bijective linear function and its inverse

Text

Given $f:\mathbb{R}\to\mathbb{R}$, $f(x)=3-2x$, show it is bijective, find the inverse and compute $f^{-1}(7)$.

Solution

Bijectivity. Being linear with slope $-2\ne 0$, $f$ is strictly decreasing (hence injective) and takes every real value (hence surjective): it is bijective. Inverse. Set $y=3-2x$ and solve $x=\dfrac{3-y}{2}$, so $f^{-1}(x)=\dfrac{3-x}{2}$. $f^{-1}(7)=\frac{3-7}{2}=-2.$ $\boxed{\,f^{-1}(x)=\dfrac{3-x}{2},\quad f^{-1}(7)=-2\,}$