Question

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

Answer 1

`x in (-oo, 0] uu (2, + oo)`

Explanation:

The first thing to look out for is any value of `x` that will maek the denominator of the fraction equal to zero. That happens when

`x -2 = 0 implies x = 2`

so this value of `x` will be excluded from the domain of the function.

The second thing to look out for is the fact that you're dealing with a square root, which, for real numbers, can only be taken of positive numbers.

This means that the fraction `x/(x-2)` must be greater than or equal to zero.

`x/(x-2) >= 0`

This condition is satisfied for

`x <= 0 implies {(x <=0), (x-2 < 0) :} implies x/(x-2) >=0`

and

`x >2 implies {( x>2), (x - 2 > 0) :} implies x/(x-2) >= 0`

Therefore, the domain of the function will include any value of `x` that is smaller than or equal to `0` or greter than `2`. In interval notation, this is equivalent to

`x in (-oo, 0] uu (2, + oo)`

graph{sqrt(x/(x-2)) [-10, 10, -5, 5]}