Question

The number of continuous functions $f$ on $R$ such that $\left(f\right(x){)}^2=x^2$ for all $x$ is

• A. $1$
• B. $2$
• C. $4$
• D. infinite

Answer 1

The correct option is C $4$

$\left(f\right(x){)}^2=x^2$
$\Rightarrow \left(f\right(x){)}^2-x^2=0$ or $\sqrt{f\left(x\right)}=\sqrt{x^2}$
$\Rightarrow f\left(x\right)=\pm x$ or $f\left(x\right)=\pm |x|$