Question:
Find the domain and the range of each of the following real
function: $f(x)=\sqrt{3 x-5}$
Solution:
Given: $f(x)=\sqrt{3 x-5}$
Need to find: Where the functions are defined.
The condition for the function to be defined,
$3 x-5 \geq 0$
$\Rightarrow x \geq \frac{5}{3}$
So, the domain of the function is the set of all the real numbers greater than equals to $\frac{5}{3}$.
The domain of the function, $\mathrm{D}_{\mathrm{f}(\mathrm{x})}=\left[\frac{5}{3}, \infty\right)$.
Putting $\frac{5}{3}$ in the function we get, $f(x)=0$
It means the range of the function is defined for all the values greater than equals to 0.
The range of the function, $\operatorname{R}_{\mathrm{f}(\mathrm{x})}=[0, \infty)$.