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Question 1051160: How would you identify the domain of 1 over cubed root x+7? or square root x-1 over 2x-3?
I want to say the cubed root is all real numbers and the square root in the numerator x cannot equal 1 or 3 but that's a guess. What's an easy way to see the domain of roots set up like this?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the domain of 1 divided by the cube root of (x + 7) would be all real values of x except x = -7.
wen x = -7, the denominator becomes 0 and the value of y is therefore undefined.
the domain of the square root of (x-1) divided by (2x-5) would be all real values of x >= 1 except x = 5/2.
when x is < 1, the expression under the square root sign is negative which results in a number that is not real and the value of y is therefore undefined.
when x equals 5/2, the denominator is equal to 0 and the value of y is therefore undefined.
when you are identifying the domain, the two main things in the equation to look for are:
the expression within the square root sign has to be greater than or equal to 0.
this also applies to the expression within any even root, such as fourth root, sixth root, etc.
the expression in the denominator cannot be equal to 0.
looking at your equations in turn, and following these instructions, you would see the following.
first equation is y = 1 / cube root of (x+7).
the cube root is an odd root so the expression within the cube root sign can be greater than or equal to 0 and can also be less than 0.
the cube root of (x+7) is in the denominator so the cube root of (x+7) can't be equal to 0.
the cube root of (x+7) is equal to 0 when x = -7.
the domain is therefore all real values of x except when x = -7.
second equation is y = square root of (x-1) divided by (2x-3).
the square root is an even root, so the expression under the square root sign has to be greater than or equal to 0.
set x-1 >= 0 and solve for x to get x >= 1.
(2x-3) is in the denominator and therefore can't be equal to 0.
set 2x-3 = 0 and solve for x to get x = 3/2.
when x = 3/2, 2x - 3 is equal to 0, therefore x can't be equal to 3/2.
your domain is all real values of x >= 1 except when x = 3/2.
the graph of y = 1 / cube root of (x+7) is shown below:
the graph produces all values of y except where x = -7.
the value of y is undefined at x = -7 because the denominator is 0 when x = -7.
the graph of y = sqrt(x-1) / (2x-3) is shown below:
the graph only shows values of y when x is greater than or equal to 1 because the value of the square root of (x-1) is undefined when x is less than 1.
the value of y is undefined at x = 3/2 because the denominator is 0 when x = 3/2.
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