First slide

Functions (XII)

Question

$Find the range of f (x) = \cot^{- 1} (2 x - x^{2}) .$

Moderate

A

$[\frac{π}{4}, π)$
B

$[\frac{π}{4}, π]$
C

$[\frac{π}{2}, π]$
D

$[- π, \frac{π}{2}]$

Solution

$\begin{array}{l} Let θ = \cot^{- 1} (2 x - x^{2}), where θ \in (0, π) \\ ⇒ \cot θ = 2 x - x^{2}, where θ \in (0, π) \\ ⇒ \cot θ = 1 - (1 - 2 x + x^{2}), where θ \in (0, π) \\ ⇒ \cot θ = 1 - (1 - x)^{2}, where θ \in (0, π) \\ ⇒ \cot θ \leq 1, where θ \in (0, π) \\ \begin{aligned} ⇒ \frac{π}{4} \leq θ < π \\ So, the range of f (x) is [\frac{π}{4}, π) . \end{aligned} \end{array}$

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