Question

The range of $f\left(x\right)=\cos \left(\sqrt{x}\right)$ is

• A. $[-1,1]$
• B. $[0,1]$
• C. $(-1,1]$
• D. $(0,1]$

Answer 1

The correct option is A $[-1,1]$

We know that, range of $\cos \theta$ is $[-1,1]$
For $f\left(x\right)=\cos \left(\sqrt{x}\right)$ to be defined $x\ge 0.$
Also, $\cos \theta \in [-1,1]$ for $\theta \ge 0$
Hence, the range of $f\left(x\right)=\cos \left(\sqrt{x}\right)$ is $[-1,1].$

Alternate solution:
We can solve this by taking different values of $x.$
$x=\frac{{\pi }^2}{4}\Rightarrow y=\cos \left(\frac{\pi }{2}\right)=0,x=0\Rightarrow y=\cos \left(0\right)=1,x={\pi }^2\Rightarrow y=\cos \left(\pi \right)=-1$

$\therefore y\in [-1,1]$