HELP!!! finding the domain for the square root of 2x/x^2-324

Question

I am struggling to understand how to find the domain of a square root with a fraction

 

sqrt2x/x^2-324 (the square root is for the entire fraction.) I get that for the denominator I need to factor, so 2x/(x+18)(x-18)≠0 and the numerator is 2x≥0. I guess I'm just lost about what I should do beyond that.

 

I'm usually really good with math, but after so many years away from it, I guess my brain has turned to complete mush. :(

 

Please help me understand this, and how to write it in interval notation

Answer 1

The square root of zero is allowed so it is not correct to say 2x ≤ 0

 

There are two major circumstances to consider the possibility that the domain of a function is not all real numbers. One is the rational function (there is a denominator) and the other is a square root (the function inside cannot be negative).

In your example we have both.

 

Let's address the denominator first. A denominator cannot be zero. Therefore x cannot be 18 or -18.

 

Now let's address the square root. The denominator changes sign at 18 and -18 because both values make the denominator zero. The denominator is positive for values > 18 and the numerator is also positive when x > 18 so we have x > 18.

 

The denominator is positive for values > -18 because both numerator and denominator are negative. However the denominator doesn't become positive until x > 18 while the numerator becomes positive when x > 0. When the numerator turns positive but the denominator remains negative, the fraction is negative. Therefore -18 < x ≤ 0

 

Interval notation    (-18, 0] and (18, ∞)

 

Answer 2

An easy way to do this one is to use a graphical calculator (e.g. TI-84) to plot the function inside the square root.

The calculator will show that   2 x/(x² - 324) is defined and non negative  in two intervals:   x > 18   and  -18 < x ≤0

The union of these two intervals is the domain of the square root function.

 

There are other approaches, but this graphical approach is the usually the best for dealing with the overall shape of rational functions such as 2x /(x² - 324)