Faith D.

asked • 08/21/12

trouble spots for the domain may occur where the denominator is ? or where the expression under a square root symbol is negative

The denominator of f(x) = sq root x-5 / x-19

the denominator is 0 when x=

2 Answers By Expert Tutors

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Rajitha A. answered • 05/23/13

Expert, experienced Math and Excel tutor.

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Olivia A. answered • 08/21/12

Engineering Student for Math, Science & Reading Tutoring

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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. 





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Giovanna C.

Case 2:

Whenever you have an irrational function, its domain is determined by the inequality obtained by placing the radicand greater than or equal to zero (=0). Therefore, the domain of this function is given by the solutions of the following inequality: [(x-5)/(x-19)]=0. Solving this algebraic inequality means to find all the x-values that makes the statement positive or equal to zero. Keep in mind that the procedure for solving this algebraic inequality excludes that the denominator is equal to zero. So, the domain will be

]-8,5] U ]19,+8[

where the value 19 doesn't belong to the domain because it makes zero the denominator.



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