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

Solution:

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

Need to find: Where the functions are defined.

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

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

Therefore,

$x=0$

It means that the denominator is zero when $\mathrm{x}=0$

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

The domain of the function, $\mathrm{D} \mathrm{f}(\mathrm{x})=(-\infty, 0) \cup(0, \infty)$.

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

So the equation becomes,

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

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

To find the range of the function $\mathrm{f}\left(\mathrm{x}_{1}\right)$ 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, $R_{f(x)}=(-\infty, 0) \cup(0, \infty)$.