SOLUTION: what is the domain of the function sqrt((10x)/(x^2-324))

Question 557350: what is the domain of the function sqrt((10x)/(x^2-324))
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
equation is:
sqrt((10x)/(x^2-324))
this formula looks like:

2 main things to consider in the domain.
the value of the square root function can't be negative.
the denominator in a division can't be equal to 0.
in your equation, the denominator is:
x^2 - 324
if the value of x^2 = 324, then the denominator will be equal to 0.
when x^2 = 324, x = +/- 18.
this means that the value of x in the domain can't be +/- 18.
if you say away from x = +/- 18, then the only thing left to consider is that the square root of the expression can't be negative.
the expression within the square root sign is:
(10x) / (x^2-324)
this expression will be be positive when:
the numerator is negative and the denominator is negative.
the numerator is positive and the denominator is positive.
the numerator will be negative when x is less than 0.
the denominator will be negative when x^2 - 324 is < 0.
we create the equation:
x^2 - 324 < 0 and solve for x.
add 324 to both sides of this equation to get:
x^2 < 324
in order for x^2 to be less than 324, the value of x must be smaller than |18|.
this means that:
x < 18 or:
x > -18
the denominator will be negative when x < 18 or x > -18.
the conditions for the numerator and denominator being both negative are:
x < 0
x < 18 or x > -18.
both these conditions are satisfied when x < 0 and x > -18
the conditions for the numerator being negative and the denominator being negative are:
-18 < x < 0 *****
now for the condition where the numerator is positive and the denominator is positive, we will need:
x >= 0 and x^2 - 324 > 0
x^2 - 324 > 0 becomes:
x^2 > 324
this condition will be satisfied when |x| > 18.
this means that:
x > 18
x < -18
since x > 0 is a condition for the numerator, then the condition for both numerator and denominator being positive are:
x > 0 and x > 18
both these conditions will be satisfied when:
x > 18 *****
we have the following conditions for the domain.
x can be any real number as long as:
x is not equal to +/- 18 *****
-18 < x < 0 *****
x > 18 *****
we can graph the equation to see if this is valid.
the equation is:
sqrt((10x)/(x^2-324))
the graph of the equation looks like:

you can see from the graph that the equation is only valid when:
-18 < x < 0
x > 18

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