Question

The mid-point of the domain of the function $f\left(x\right)=\sqrt{4-\sqrt{2x+5}}$ for real $x$ is

• A. $\frac{1}{4}$
• B. $\frac{3}{2}$
• C. $\frac{2}{3}$
• D. $-\frac{2}{5}$

Answer 1

The correct option is B $\frac{3}{2}$

The term inside the square root has to be non-negative.

$\Rightarrow 4-\sqrt{2x+5}\ge 0$

$\therefore 4\ge \sqrt{2x+5}$

Squaring both sides, $16\ge 2x+5$

$\therefore 2x\le 11$ or $x\le 5.5$

Also, since the term $2x+5$ is also inside a square root, $2x+5\ge 0$

i.e. $x\ge -2.5$

The domain of the function therefore becomes $[-2.5,5.5]$

Midpoint of the domain is $\frac{-2.5+5.5}{2}=1.5=\frac{3}{2}$