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Problem 35 Hard Difficulty

Find the domain of the function.

$ h(x) = \dfrac{1}{\sqrt[4]{x^2 - 5x}} $

Answer

Domain $=(-\infty, 0) \cup(5, \infty)$

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Video Transcript

to find the domain of this function. There are a couple of things we want to think about. First of all, because we have 1/4 root, you have to have something that is non negative inside of four through. If you want to get a real result, so you need to have either zero or positive. And because we have a denominator in this problem, we need the denominator to not be zero. So actually, we need what's inside the fourth route to just be positive. So we need X squared minus five x to be greater than zero. So how do we solve that inequality? We call that a quadratic inequality or a non linear inequality. It's a little bit more complicated than solving a plane. Linear inequality and the way I would start is by factoring X out of both terms. So if X Times X minus five is greater than zero, so notice that we have a product of two things is greater than zero. How can that happen? That can happen if you're multiplying, a positive and a positive, or if you're multiplying and negative and a negative. So we have two cases to consider one is if X is greater than zero and X minus five is greater than zero. The other one is if X is less than zero and X minus five is less than zero, So two positives multiply to give you a positive two negatives multiply. To give you a positive, it has to be one or the other. Okay, so let's sell both of these. So we have X is greater than zero and X is greater than five. OK, if you put those two ideas together, the only way for both of those to be true at the same time is if X is greater than five. Now we look at the other two. Inequalities X is less than zero and X is less than five. The only way for those two things to be true at the same time is if X is less than zero. So putting all of this together exes let X is greater than five or X is less than zero, and we can write it like this if you want to use interval notation than the domain will be negative. Infinity 20 Union five to infinity or, if you want to use inequality notation, you can say X is less than zero or X is greater than five