Question

What is the range of the function $f\left(x\right)=\frac{|x-1|}{x-1}$?

Answer 1

$f\left(x\right)=\frac{|x-1|}{x-1}$

For $x<1:$

For $x<1$, $|x-1|$ could be written as $-(x-1)$. Then,

$\Rightarrow$ $f\left(x\right)=\frac{-(x-1)}{x-1}=-1$

For $x>1:$

For $x>1$, $|x-1|$ could be written as $(x-1)$. Then,

$\Rightarrow$ $f\left(x\right)=\frac{(x-1)}{x-1}=1$

Hence, range of the function $=\{-1,1\}$