Question

If $f\left(x\right)=\sqrt{2-x}\text{and}g\left(x\right)=\sqrt{1-2x}$, then the domain of $f\left(g\right(x\left)\right)$ is.

• A. $(-\infty ,\frac{1}{2}]$
• B. $(\frac{1}{2},\infty ]$
• C. $(-\infty ,\frac{3}{2}]$
• D. $[-\frac{3}{2},\frac{1}{2}]$

Answer 1

The correct option is D $[-\frac{3}{2},\frac{1}{2}]$

$f\left(g\right(x\left)\right)=\sqrt{2-\sqrt{1-2x}}$

For given function to be defined,

$1-2x\ge 0\Rightarrow x\le \frac{1}{2}$

And $2-\sqrt{1-2x}\ge 0\Rightarrow \sqrt{1-2x}\le 2$

$\Rightarrow 1-2x\le 4\Rightarrow x\ge -\frac{3}{2}$

Thus required domain is $[-\frac{3}{2},\frac{1}{2}]$