First slide

Functions (XII)

Question

$The range of the function, f (x) = \cot^{- 1} \log_{0.5} (x^{4} - 2 x^{2} + 3) i s$

Moderate

A

$(0, π)$
B

$(0, 3 π / 4]$
C

$[3 π / 4, π)$
D

$[π / 2, 3 π / 4]$

Solution

$x^{4} - 2 x^{2} + 3 = {(x^{2} - 1)}^{2} + 2 \geq 2 \Rightarrow \log {}_{0.5}{(x^{4} - 2 x^{2} + 3)} \leq \log {}_{0.5}2 = - \log {}_{2}2 = - 1 (\sin c e x \geq y \Rightarrow \log {}_{a}x \leq \log {}_{a}y i f 0 < a < 1)$
$\Rightarrow c o t^{- 1} (\log {}_{0.5}{(x^{4} - 2 x^{2} + 3)}) \geq c o t^{- 1} (- 1) (\sin c e c o t^{- 1} x f u m c t i o n i s d e c r e a \sin g)$
$\Rightarrow y = c o t^{- 1} (\log {}_{0.5}{(x^{4} - 2 x^{2} + 3)}) \geq π - \cot^{- 1} 1 = \frac{3 π}{4} \Rightarrow y \in [\frac{3 π}{4}, π) (since \cot^{- 1} x \in (0, π))$

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