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

Given: $f(x)=\frac{1}{(x-5)}$

Need to find: Where the functions are defined.

Let, $f(x)=\frac{1}{x-5}=y$ ........(1)

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

Therefore,

$x-5=0$

$\Rightarrow x=5$

It means that the denominator is zero when $x=5$

So, the domain of the function is the set of all the real numbers except 5 .

The domain of the function, $\mathrm{D}_{\mathrm{f}(\mathrm{x})}=(-\infty, 5) \cup(5, \infty)$.

Now, to find the range of the function we need to interchange $x$ and $y$ in the equation no. (1)

So the equation becomes,

$\frac{1}{y-5}=x$

$\Rightarrow y-5=\frac{1}{x}$

$\Rightarrow y=\frac{1}{x}+5=\frac{1+5 x}{x}=f\left(x_{1}\right)$

To find the range of the function $f\left(x_{1}\right)$ we need to equate the denominator of the function to 0 .

Therefore,

x = 0

It means that the denominator is zero when $x=0$

So, the range of the function is the set of all the real numbers except 0 .

The range of the function, $\operatorname{Rf}(x)=(-\infty, 0) \cup(0, \infty)$.