Question

Which of the following statements is incorrect?

• A. $xsgnx=\left|x\right|$
• B. $\left|x\right|sgnx=x$
• C. $x(sgnx)(sgnx)=x$
• D. $\left|x\right|{(sgnx)}^3=\left|x\right|$

Answer 1

The correct option is D

$\left|x\right|{(sgnx)}^3=\left|x\right|$

Explanation for the correct option

For option (d)

When $x>0$

$\left|x\right|{(sgnx)}^3=x\times {(+1)}^3$

$$\begin{gathered} =x\times 1 \\ =x \\ =\left|x\right| \\ =R.H.S. \end{gathered}$$

When $x=0$

$\left|x\right|{(sgnx)}^3=0\times {\left(0\right)}^3$

$$\begin{gathered} =0 \\ =\left|x\right| \\ =R.H.S. \end{gathered}$$

When $x<0$

$\left|x\right|{(sgnx)}^3=x\times {(-1)}^3$

$$\begin{gathered} =-x \\ \ne R.H.S. \end{gathered}$$

For all values of $x$, $\left|x\right|{(sgnx)}^3=\left|x\right|$ is not satisfied.

Explanation for incorrect options

For option (a)

When $x>0$

$xsgnx=x\times (+1)$

$$\begin{gathered} =x\times 1 \\ =x \\ =\left|x\right| \\ =R.H.S. \end{gathered}$$

When $x=0$

$xsgnx=0\times \left(0\right)$

$$\begin{gathered} =0 \\ =\left|x\right| \\ =R.H.S. \end{gathered}$$

When $x<0$

$xsgnx=-x\times (-1)$

$$\begin{gathered} =x \\ =\left|x\right| \\ \ne R.H.S. \end{gathered}$$

For all values of $x$, $xsgnx=\left|x\right|$ is satisfied.

For option (b)

When $x>0$

$\left|x\right|sgnx=x\times (+1)$

$$\begin{gathered} =x\times 1 \\ =x \\ =R.H.S. \end{gathered}$$

When $x=0$

$\left|x\right|sgnx=0\times \left(0\right)$

$$\begin{gathered} =0 \\ =R.H.S. \end{gathered}$$

When $x<0$

$\left|x\right|sgnx=x\times (-1)$

$$\begin{gathered} =-x \\ =R.H.S. \end{gathered}$$

For all values of $x$, $\left|x\right|sgnx=x$ is satisfied.

For option (c)

When $x>0$

$x(sgnx)(sgnx)=x\times (+1)(+1)$

$$\begin{gathered} =x\times 1 \\ =x \\ =R.H.S. \end{gathered}$$

When $x=0$

$x(sgnx)(sgnx)=0\times \left(0\right)\times \left(0\right)$

$$\begin{gathered} =0 \\ =R.H.S. \end{gathered}$$

When $x<0$

$x(sgnx)(sgnx)=-x\times (-1)(-1)$

$$\begin{gathered} =-x \\ =R.H.S. \end{gathered}$$

For all values of $x$, $x(sgnx)(sgnx)=x$ is not satisfied.

Hence, the correct option is (D).