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Given that $f(y)=\frac{ \mid y \mid }{y},$ and $q$ is non-zero real number, the value of $\mid f(q)-f(-q) \mid$ is

1.  $0$
2.  $-1$
3.  $1$
4. $2$

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0
It will be 2.

$f(y) = \frac{|y|}{y}= \left\{\begin{matrix} +1 & when \; y>0\\ -1 & when \; y<0 \end{matrix}\right.$

It is well-known signum function but in the given quetion, it is not defined for $y=0$

Here, $q$ is a non-zero real number.

So,

$(1)$ When $q>0$, then $|f(q)-f(-q)| = |(+1) - (-1)| = 2$ because when $q>0$ then $f(q)=f(>0)=+1$ and $q>0$ means $-q<0$ which implies  $f(-q)=f(<0)=-1$

$(2)$ When $q<0$, then $|f(q)-f(-q)| = |(-1) - (+1)| = 2$ because when $q<0$ then $f(q)=f(<0)=-1$ and $q<0$ means $-q>0$ which implies  $f(-q)=f(>0)=+1$

Correct Option: D.

by Boss (14k points)
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