Question

How do you find the domain of `f(x) = sqrt((x+3)/(2-x))`?

Answer 1

Domain: `-3 <= x < 2`, in interval notation : `[-3 , 2)`

Explanation:

`f(x) = sqrt((x+3)/(2-x))`

Domain: Restriction `x -2 != 0 or x !=2` , Under root must not be

negative quantity , so `(x+3)/(2-x) >=0` , critical points are

`x= -3 and x= 2`
Sign change:

When `x < -3` sign of `(x+3)/(2-x)` is `(-)/(+)= (-) ; < 0`

When `-3 <= x < 2` sign of `(x+3)/(2-x)` is `(+)/(+)= (+) ; >= 0`

When `x >2` sign of `(x+3)/(2-x)` is `(+)/(-)= (-) ; < 0`

so for `-3 <= x < 2 , (x+3)/(2-x) >=0`

Domain: `-3 <= x < 2`, in interval notation : `[-3 , 2)`

graph{((x+3)/(2-x))^0.5 [-11.25, 11.25, -5.625, 5.625]} [Ans]