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

Checking the options
$(a) LHS = x sgn x = x \times (+ 1) = x if x > 0$
$\begin{array}{l} x \times 0 = 0 if x = 0 \\ \begin{array}{ll} x \times (- 1) & = - x if x < 0 \\ = | x | = RHS \end{array} \end{array}$
Therefore, statement is correct.
$(b) LHS = | x | sgn x = (+ x) \times (+ 1) = x if x > 0$
$\begin{array}{lc} 0 \times 0 = 0 & if x = 0 \\ (- x) \times (- 1) = x & if x < 0 \end{array}$
Therefore, statement is correct.
$(c) LHS = (x (sgn x)) \times sgn x = | x | sgn x$
$\begin{aligned} = x \\ = RHS \end{aligned}$
Therefore, statement is correct.
$(d) LHS = (| x | sgn x) (sgn x)^{2} = x (sgn x)^{2}$
$\begin{array}{l} = | x | \times (sgn x) \\ = x \\ \neq RHS \end{array}$
Therefore, statement is incorrect.

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