First slide

Functions (XII)

Question

Which of the following statements is incorrect?

Moderate

A

$x sgn x = | x |$
B

$| x | sgn x = x$
C

$x (sgn x) (sgn x) = x$
D

$| x | {(sgn x)}^{3} = x$

Solution

$| x | =$ ${\begin{matrix} - x; x < 0 \\ 0; x = 0 \\ x; x > 0 \end{matrix}$ $a n d sgn (x)$ $= {\begin{matrix} - 1; x < 0 \\ 0; x = 0 \\ 1; x > 0 \end{matrix}$
$∴$ (A) $x sgn x = {\begin{matrix} - x, i f x < 0 \\ 0, i f x = 0 \\ x, i f x > 0 \end{matrix} \Rightarrow$ $x sgn x = | x |$

(B) $| x | sgn x = (- x) (- 1) = x,$       if                        $x < 0$

                   $(0) (0) = 0,$                      if                        $x = 0$

                   $(x) (1) = x,$                        if                         $x > 0$

$\Rightarrow | x | sgn x = x$

(C) $x (sgn x) (sgn x) = x$

(D) $| x | {(sgn x)}^{3} = (- x) {(- 1)}^{3} = x$   if   $x < 0$

                     $(0) {(0)}^{3} = 0,$                      if                         $x = 0$

                     $(x) {(1)}^{3} = x,$                        if                        $x > 0$

$∴ | x | {(sgn x)}^{3} = x n o t | x |$ .

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