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

Given: $f(x)=\frac{x^{2}-16}{x-4}$

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, $\mathrm{Df}(x)=(-\infty, 4) \cup(4, \infty)$.

Now if we put any value of $x$ from the domain set the output value will be either (-ve) or (+ve), but the value will never be 8

So, the range of the function is the set of all the real numbers except 8.

The range of the function, $\mathrm{R}_{\mathrm{f}(x)}=(-\infty, 8) \cup(8, \infty)$.