Rajitha A. answered • 05/23/13

Expert, experienced Math and Excel tutor.

**Case: 1) f(x) =sqrt(x-5)/(x-19) (If square root is only for numerator)**

**ANSWER:** The function F tells us to take the square root of x-5 and divide this result by x-19.

This requires that for the numerator => x-5≥0,so x≥5 , and also that for the denominator x-19≠0, so x≠19.

Combining these two restrictions, the domain of F is [5,19)U(19,infinity)

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**Case: 2)**** f(x)=sqrt(x-5/x-19) (If the square root is for both numerator and denominator)**

**ANSWER:**

The function F tells us to take the square root of x-5 and divide this result by square root of x-19.

This requires that for the numerator => x-5≥0,so x≥5 , and also that for the denominator x-19>0, so x>19.

Combining these two restrictions, the domain of F is (19,infinity)

Olivia A.

I may have interpreted this question wrong as it seems that the denominator is the whole expression sqrt(x-5/x-19). In that case, the domain is determined by what wil make the expression 0 on the denominator, as well as negative. First, again, we set x - 19 = 0. However, it gets a little more tricky when we try to see what makes this function negative.

For example, when you plug in 6, for 'x', the expression is negative as you have 1/(-13). But when you plug in a number like 4, 4 - 5 produces -1 while 4-19 produces -15 where as -1/-15 is going to give you a positive number. Therefore, by setting x -5 = 0, we find another critical point. Therefore our domain changes a little. Numbers smaller than 5 (even negative numbers) will ultimately give us a positive number under the square root while numbers in between 5 and 19 give us a negative number under the square root. Therefore your new domain is expressed in interval notation below.

(-infinity,5)U(19,infinity)

08/22/12