First slide

Functions (XII)

Question

$The range of the function f (x) = \tan \sqrt{\frac{π^{2}}{9} - x^{2}} is$

Moderate

A

$[0, \sqrt{3}]$
B

$(0, \sqrt{3})$
C

$[0, \sqrt{3})$
D

$[0, \frac{1}{\sqrt{3}}]$

Solution

$For f (x) to be defined, \frac{π^{2}}{9} - x^{2} \geq 0$
$\begin{aligned} \Rightarrow - \frac{π}{3} \leq x \leq \frac{π}{3} \\ Domain of f = [- \frac{π}{3}, \frac{π}{3}] \end{aligned}$
The greatest value of
$\begin{aligned} f (x) = \tan \sqrt{\frac{π^{2}}{9} - 0} = \sqrt{3} when x = 0 \end{aligned}$
and the least value of
$\begin{aligned} f (x) = \tan \sqrt{\frac{π^{2}}{9} - \frac{π^{2}}{9}} = 0 when x = \frac{π}{3} \end{aligned}$
$Range of function = [0, \sqrt{3}]$

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