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Increasing and Decreasing Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions2 Marks
Question
If g (x) is a decreasing function on R and $\text{f}(\text{x})=\tan^{-1}[\text{g}(\text{x})].$ State whether f(x) is increasing or decreasing on R.
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Answer

Given: g(x) is decreasing on R.
$\Rightarrow\text{x}_1<\text{x}_2$
$\Rightarrow\text{g}(\text{x}_1)>\text{g}(\text{x}_2)$
Applying $\tan^{-1}$ on both sides we get,
$\Rightarrow\tan^{-1}\{\text{g}(\text{x}_1)\}>\tan^{-1}\{\text{g}(\text{x}_2)\}$
$\Rightarrow\text{f}(\text{x}_1)>\text{f}(\text{x}_2)$

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