First slide

Functions (XII)

Question

The range of $f (x) = \sin^{- 1} (\sqrt{x^{2} + x + 1}) is$

Moderate

A

$(0, \frac{π}{2}]$
B

$(0, \frac{π}{3}]$
C

$[\frac{π}{3}, \frac{π}{2}]$
D

$[\frac{π}{6}, \frac{π}{3}]$

Solution

For the function to get defined, $0 \leq x^{2} + x + 1 \leq 1$ , but
$x^{2} + x + 1 \geq \frac{3}{4} or \frac{\sqrt{3}}{2} \leq \sqrt{x^{2} + x + 1} \leq 1$
or $\frac{π}{3} \leq \sin^{- 1} (\sqrt{x^{2} + x + 1}) \leq \frac{π}{2}$

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