First slide

Functions (XII)

Question

The domain of the function $f (x) = \sqrt{\cos^{- 1} (\frac{1 - | x |}{2})} is$

Moderate

A

$(- \infty, - 3) \cup (3, \infty)$
B

$[- 3, 3]$
C

$(- \infty, - 3] \cup [3, \infty)$
D

$ϕ$

Solution

$\cos^{- 1} (\frac{1 - | x |}{2}) is defined if - 1 \leq \frac{1 - | x |}{2} \leq 1 \begin{array}{l} \Rightarrow - 2 \leq 1 - | x | \leq 2 \Rightarrow - 3 \leq - | x | \leq 1 \\ \Rightarrow - 1 \leq | x | \leq 3 \end{array}$
|x| ≥ – 1 is true for all real values of x.
$Also \sqrt{\cos^{- 1} (\frac{1 - | x |}{2})} is defined if \cos^{- 1} (\frac{1 - | x |}{2}) \geq 0 \begin{array}{l} \Rightarrow \frac{1 - | x |}{2} \geq \cos 0 = 1 \\ \Rightarrow 1 - | x | \geq 2 \\ \Rightarrow | x | \leq - 1 (Absurd) \end{array}$
Domain ( f ) = [– 3, 3]

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