Question

If $f\left(x\right)=x|x|$ then $f^{-1}\left(x\right)$ equals-

• A. $\sqrt{|x|}$
• B. $\left(sgnx\right).\sqrt{|x|}$
• C. $-\sqrt{|x|}$
• D. Does not exist

Answer 1

Answer: (A)

$\sqrt{|x|}$

Given, $f\left(x\right)=x|x|$.

Let $y=f\left(x\right)$.

Now for $x<0$, $y=-x^2$ then $x=\sqrt{-y}$ for $y<0$.....(1).

Again for $x\ge 0$, $y=x^2$ then $x=\sqrt{y}$ for $y\ge 0$.....(2).

Now from (1) and (2) we get,

$x=\sqrt{|y|}$ for $y\in R$.