Show a function is monotonic and find a formula for its inverse

by RoRi

September 5, 2015

Let $f(x) = 1-x$ . Show that $f$ is strictly monotonic on $\mathbb{R}$ . Find the domain of the inverse of $f$ , denoted by $g$ . Find a formula for computing $g(y)$ for each $y$ in the domain of $g$ .

First, to show $f$ is monotonic let $x_1, x_2 \in \mathbb{R}$ with $x_1 < x_2$ . Then

$x_1 < x_2 \quad \implies \quad -x_1 > -x_2 \quad \implies \quad 1-x_1 > 1-x_2 \quad \implies \quad f(x_1) > f(x_2).$

Hence, $f$ is strictly decreasing on $\mathbb{R}$ .

Next,

$\begin{align*} y = 1-x && \implies && x &= 1-y \\ && \implies && g(y) &= 1-y \qquad \text{for all } y \in \mathbb{R}. \end{align*}$

Stumbling Robot

Show a function is monotonic and find a formula for its inverse

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