**Question:**

Find the domain and the range of each of the following real

function: $f(x)=\frac{|x-4|}{x-4}$

**Solution:**

Given: $f(x)=\frac{|x-4|}{x-4}$

Need to find: Where the functions are defined.

To find the domain of the function f(x) we need to equate the denominator of the function to 0.

Therefore,

$x-4=0$

$\Rightarrow x=4$

It means that the denominator is zero when $x=4$

So, the domain of the function is the set of all the real numbers except 4 .

The domain of the function, $\mathrm{D} \mathrm{f}(\mathrm{x})=(-\infty, 4) \cup(4, \infty)$.

The numerator is an absolute function of the denominator. So, for any value of $x$ from the domain set, we always get either $+1$ or $-1$ as the output. So, the range of the function is a set containing $-1$ and $+1$

Therefore, the range of the function, $R_{f(x)}=\{-1,1\}$