What is the range of the function $f(x)=\frac{|x|}{x},x\ne 0$?

The function is $f(x)=\frac{|x|}{x}$ for $x\ne 0$.We define $|x|$ as:
$$|x|=\{\begin{array}{ll}x,& \text{if }x>0\\ -x,& \text{if }x<0\end{array}$$
For $x>0$:
$$f(x)=\frac{x}{x}=1$$
For $x<0$:
$$f(x)=\frac{-x}{x}=-1$$
Since the function is only defined for $x\ne 0$, the only possible output values are $1$ and $-1$.Thus, the range is $\{-1,1\}$.