Question

If g (x) is a decreasing function on R and f(x) = tan⁻¹ [g (x)]. State whether f(x) is increasing or decreasing on R.

Answer 1

$$\begin{gathered} \text{Given:}g\left(x\right)\text{is decreasing on}\mathit{\text{R}}\text{.} \\ \Rightarrow x_1<x_2 \\ \Rightarrow g\left(x_1\right)>g\left(x_2\right) \\ {\text{Applying tan}}^{-1}\text{on both sides, we get} \\ \Rightarrow {\tan }^{-1}\left\{g\left(x_1\right)\right\}>{\tan }^{-1}\left\{g\left(x_2\right)\right\} \\ \Rightarrow f\left(x_1\right)>f\left(x_2\right) \\ \text{Thus,} \\ {x}_1<x_2\Rightarrow f\left(x_1\right)>f\left(x_2\right) \\ \text{So,}f\left(x\right)\text{is decreasing on}\mathit{\text{R.}} \end{gathered}$$