Question

Show that f(x) = $\frac{1}{x}$ is a decreasing function on (0, ∞).

Answer 1

$$\begin{gathered} \mathrm{Here}, \\ f\left(x\right)=\frac{1}{x} \\ \text{Let}{x}_1,x_2\in \left(0,\infty \right)\text{such that}{x}_1<x_2.\mathrm{Then}, \\ {x}_1<x_2 \\ \Rightarrow \frac{1}{x_1}>\frac{1}{x_2} \\ \Rightarrow f\left(x_1\right)>f\left(x_2\right) \\ \text{∴}{x}_1<x_2 \\ \Rightarrow f\left(x_1\right)>f\left(x_2\right),\forall x_1,x_2\in \left(0,\infty \right) \\ \text{So,}f\left(x\right)\text{is decreasing on}\left(0,\infty \right). \end{gathered}$$