Question

What is the range of $f\left(x\right)=\tan \left(\sqrt{\frac{{\pi }^2}{4}-x^2}\right)$ ?

• A. $(-\infty ,\infty )$
• B. $(0,\infty )$
• C. $[0,\infty )$
• D. $(-\infty ,0)$

Answer 1

The correct option is C $[0,\infty )$

$f\left(x\right)=\tan \left(\sqrt{\frac{{\pi }^2}{4}-x^2}\right)$
$\frac{{\pi }^2}{4}-x^2>0$
$\Rightarrow x^2\le \frac{{\pi }^2}{4}$
$\Rightarrow \frac{-\pi }{2}\le x\le \frac{\pi }{2}$

When $x=\pm \frac{\pi }{2}\Rightarrow f\left(x\right)=\tan 0=0$
When $x=0\Rightarrow f\left(x\right)=\tan \frac{\pi }{2}=\infty$
Therefore, the range is $[0,\infty )$