Question

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

• A. $[0,3]$
• B. $[0,\sqrt{3}]$
• C. $[3,\sqrt{3}]$
• D. $[\sqrt{3},3]$

Answer 1

The correct option is A $[0,3]$

$f\left(x\right)=\tan \sqrt{\frac{{\pi }^2}{9}-x^2}$

${\tan }^{-1}y=\frac{{\pi }^2}{9}-x⟹x^2=\frac{{\pi }^2}{9}-{\tan }^{-1}y⟹x=\pm \sqrt{\frac{{\pi }^2}{9}-{\tan }^{-1}y}$

$f^{-1}\left(x\right)=\pm \sqrt{\frac{{\pi }^2}{9}-{\tan }^{-1}y}$

$\therefore y\in [0,3].$