First slide

Introduction to ITF

Question

The Range of $\tan^{- 1} (\frac{2 x}{1 + x^{2}})$ is

Easy

A

$[- \frac{π}{4}, \frac{π}{4}]$
B

$(- \frac{π}{2}, \frac{π}{2})$
C

$(- \frac{π}{2}, \frac{π}{4}]$
D

$[\frac{π}{4}, \frac{π}{2}]$

Solution

$W e h a v e 1 + x^{2} \geq 2 | x | \Rightarrow \frac{2 | x |}{1 + x^{2}} \leq 1$
$\begin{array}{l} \Rightarrow - 1 \leq \frac{2 x}{1 + x^{2}} \leq 1 \end{array} \Rightarrow \tan^{- 1} (- 1) \leq \tan^{- 1} (\frac{2 x}{1 + x^{2}}) \leq \tan^{- 1} (1) (∵ \tan^{- 1} i s i n c r e a \sin g) \Rightarrow - \frac{π}{4} \leq \tan^{- 1} (\frac{2 x}{1 + x^{2}}) \leq \frac{π}{4}$

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