GATE2011 AG: GA-7

Question

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

• A.  $0$
• B.  $-1$ 
• C.  $1$       
• D. $2$

Comments

It will be 2.

---

Answer will be 2

Answer 1

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

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.

Comments

In this question we have to consider 2 cases because they mentioned q is non zero so q can be positive or negative but not zero i.e  

When q>0 and q<0

 

Case-1: when q>0  

f(y) = |y| / y  

f(q) = |+q| / +q =1

f(-q)=|-q| / -q  = -1  

So |f(q) - f(-q)|  

= |1 - (-1)|  

= 2  

Case-2:  

When q<0  

As q<0  

f(q) = |-q| / -q =-1  

f(-q)=|q| /q =1  

So |f(q) - f(-q)|  

=|-1 – 1|

= 2

so correct answer is option – D