First slide

Functions (XII)

Question

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

Easy

A

$[- \frac{π}{4}, \frac{π}{4}]$
B

$(- π, π)$
C

$[- \frac{π}{4}, \frac{π}{4})$
D

$(- \frac{π}{2}, \frac{3 π}{4}]$

Solution

First, we must get the range of
$\frac{2 x}{1 + x^{2}} = y$
$\begin{array}{l} We have y x^{2} - 2 x + y = 0 \\ Since x is real, D \geq 0, i.e., 4 - 4 y^{2} \geq 0 or - 1 \leq y \leq 1 . So, \\ \tan^{- 1} (y) \in [- \frac{π}{4}, \frac{π}{4}] (As \tan x is an increasing function) \end{array}$

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