Question

In the following exercises, find the domain of the function and write the domain in interval notation.

$F\left(x\right)=\sqrt{\frac{x+3}{x-2}}$

Short answer

Domain is$x\in (-\infty ,-3]\cup (2,\infty )$

Step-by-step solution

Given information 

Domain of$F\left(x\right)=\sqrt{\frac{x+3}{x-2}}$

Solution

We have

$F\left(x\right)=\sqrt{\frac{x+3}{x-2}}$

Here

$\frac{x+3}{x-2}\ge 0$

Taking the intervals
$(-\infty ,-3],(-3,2],(2,\infty )$

Checking

In localid="1663766527461" $(-\infty ,-3]$value of $\frac{x+3}{x-2}\ge 0$

In localid="1663766706310" $(-3,2]$value of localid="1663766578080" $\frac{x+3}{x-2}<0$is not defined or negative

In localid="1663766538327" $(2,\infty )$value of $\frac{x+3}{x-2}\ge 0$

Hence the domain is

$x\in (-\infty ,-3]\cup (2,\infty )$