The range of the function

$f(x)=\frac{x+2}{|x+2|}$

$= \begin{cases}\frac{x+2}{x+2} & ; x \geq-2 \\ \frac{x+2}{-(x+2)} & ; x<-2\end{cases}$

i. e $f(x)= \begin{cases}1 & ; x \geq-2 \\ -1 & ; x<-2\end{cases}$

$\therefore$ Range of function $f(x)$ is $\{-1,1\}$