Question

The domain of the function $f\left(x\right)=\sqrt{4x-3}+\sqrt{2x-6}$ is

• A. $[0,\infty )$
• B. $[\frac{3}{4},\infty )$
• C. $[\frac{4}{3},\infty )$
• D. $[3,\infty )$

Answer 1

The correct option is D $[3,\infty )$

$f\left(x\right)=\sqrt{4x-3}+\sqrt{2x-6}$

For function to be defined we need to have

$4x-3⩾0$ and $2x-6⩾0$ together.

$\begin{array}{cc}\text{Case}I\to & 4x-3⩾0\\ & \Rightarrow x⩾\frac{3}{4}\\ & \therefore x\in \left[\frac{3}{4}\infty \right)\end{array}$

$\begin{array}{cc}\text{Case}I\to & 2x-6⩾0\\ & x⩾3\\ & \therefore x\in \left[3\infty \right)\end{array}$

we need intersection of values of $x$ in both cases

so finally $x\in \left[3\infty \right)$

Answer option (D).