Question

The domain of the function, $f\left(x\right)=\sqrt{2-x}$$-\frac{1}{\sqrt{9-x^2}}$ is

• A. (-3,1)
• B. [-3,1]
• C. (-3,2)
• D. (-3,2]

Answer 1

The correct option is D (-3,2]

Given,

$f\left(x\right)=\sqrt{2-x}-\frac{1}{\sqrt{9-x^2}}$

$\sqrt{f\left(x\right)}\Rightarrow f\left(x\right)\ge 0$

$2-x\ge 0:x\le 2$

$9-x^2\ge 0:-3\le x\le 3$

Combined interval:

$-3\le x\le 2$

Undefined singularity points: $x=3,x=-3$

Combining real regions and undefined singularity points:

$-3<x\le 2$

$(-3,2]$