Question 405963: determine the domain:
h(y) = (2y +9)/sqrt(y + 5)
please help! i had to rewrite the question it was misinterpreted.
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! 
Finding the domain of a function is a process of determining what set of numbers can be used as input for the function. For some functions some numbers cannot be used because they would violate some basic Mathematical principle. As you learn more Math the list of things to avoid slowly grows. Here's a partial list which includes the ones we need for this problem,- Division by zero is undefined. So denominators must never be zero. If a function has denominators with a variable in them, then the variable must not be allowed to be a number that would cause any denominator to becomes zero.
- Raising a real number to an even power, like squaring, never results in a negative number. Raising to an even power always results in a positive number or zero. Since the radicand (the expression inside a radical) of an even-numbered root represents the result of raising a number to an even power, it must never be negative. If a variable is in the radicand of an even-numbered root, like square root, then it must not cause the radicand to become negative.
- Negative or zero bases and arguments of logarithms are undefined. So a variable in either of those locations must not be allowed to have a value that would cause the base or the argument of any logarithm to become zero or negative.
Often finding domains is done by finding all the numbers the input variable cannot be. Then the domain is all real numbers except those.
Your function has a denominator with a variable in it. So we must make sure the value of the variable never causes the denominator to become zero. To find out what number(s), if any, that would make the denominator zero we solve the equation:
denominator = 0
In this function we solve:
Squaring both sides we get:
y+5 = 0
Subtracting 5 from each side we get:
y = -5
So y cannot be -5 because it would make the denominator zero.
Your has an even numbered root, the square root. And there is a variable in its radicand. So we must make sure the value of variable never makes the radicand negative. To find what values the input variable, y, cannot be we solve the inequality:
radicand < 0
(To find what numbers y can be we solve  .)
For your function this inequality is:
y+5 < 0
To solve this we just subtract 5 from each side:
y < -5
So we have found that y cannot be -5 because it would make the denominator zero and y cannot be less than -5 because that would make the radicand of a square root negative. y can be any number except these. So the domain of h(y) is all real numbers except these. IOW, the domain is all number greater than -5. In interval notation this is: (-5,  )
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