Find the domain and the range of each of the following real function: f(x)
$=\frac{1}{\sqrt{2 x-3}}$
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, $\operatorname{Rf}(x)=(0,1)$.
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