Question

Range of $f\left(x\right)=\frac{x}{|x|}is$ :

• A. $R-\left(0\right)$
• B. $R-(-1,1)$
• C. $-1,1$
• D. None of these

Answer 1

Answer: (C)

$-1,1$

$Case1:$

In the first case, if $x$ is positive,

And since the modulus of a positive number is the number itself, our function becomes,

$f\left(x\right)=\frac{x}{x}=1,$

here I have replaced modulus of $x,$ by $x,$ because there is no change.

$Case2:$

In the second case, when $x$ is negative, the value of our function changes to,

$f\left(x\right)=\frac{-x}{x}=-1.$

Because the numerator stays the same, however our denominator has the modulus function which turns it into its positive counterpart. So while the numerator is $(-x)$, our denominator has already become $(+x).$

$Case3:$

And the third case is invalid, because $\frac{0}{0}$ is not defined.

Hence our domain becomes $(-\infty ,\infty )-\left\{0\right\}$

And our range would be, the only two possible solutions, $\{-1,1\}.$