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

Given: $f(x)=\frac{1}{\sqrt{2 x-3}}$

Need to find: Where the functions are defined.

Let, $f(x)=\frac{1}{\sqrt{2 x-3}}=y$ .......(1)

The condition for the function to be defined,

$2 x-3>0$

$\Rightarrow x>\frac{3}{2}$

So, the domain of the function is the set of all the real numbers greater than $\frac{3}{2}$.

The domain of the function, $\mathrm{D}_{\mathrm{f}(\mathrm{x})}=\left(\frac{3}{2}, \infty\right)$.

Now putting any value of x within the domain set we get the value of the function always a fraction whose denominator is not equals to 0.

The range of the function, $\operatorname{Rf}(x)=(0,1)$.