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

1 Answer
Stefan V.
Oct 13, 2015

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]}

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