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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 ?
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Solution
\[\text { Given }:g\left( x \right)\text { is decreasing on R }.\]
\[ \Rightarrow x_1 < x_2 \]
\[ \Rightarrow g\left( x_1 \right) > g\left( x_2 \right)\]
\[ \text {Applying tan}^{- 1} \text { on both sides, we get }\]
\[ \Rightarrow \tan^{- 1} \left\{ g\left( x_1 \right) \right\} > \tan^{- 1} \left\{ g\left( x_2 \right) \right\}\]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right)\]
\[\text { Thus },\]
\[ x_1 < x_2 \Rightarrow f\left( x_1 \right) > f\left( x_2 \right)\]
\[\text { So,}f\left( x \right)\text { is decreasing on R }.\]
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