Find the Range of \( f(x)=\frac{|x|}{x} \)
Question:
Let
\[ A=\{x\in R:x\ne0,\,-4\le x\le4\} \]
and
\[ f:A\to R,\qquad f(x)=\frac{|x|}{x} \]
for \(x\in A\). Then the range of \(f\) is
(a) \(\{1,-1\}\)
(b) \(\{x:0\le x\le4\}\)
(c) \(\{1\}\)
(d) \(\{x:-4\le x\le0\}\)
Solution:
If \(x>0\),
\[ \frac{|x|}{x}=1 \]
If \(x<0\),
\[ \frac{|x|}{x}=-1 \]
Therefore, possible values of \(f(x)\) are
\[ \{-1,1\} \]
\[ \boxed{\text{Correct Answer: (a)}} \]