In Exercises 76–81, find the domain of each function. f(x) = x/(x...

Question

In Exercises 76–81, find the domain of each function. f(x) = x/(x^2 + 4x -21)

Accepted Answer

Welcome back. I'm so glad you're here. We've got a function that is H. Of X equals five X divided by X squared minus X minus 30 and we want to find its domain. So what do we know about a fraction? Well we know that the denominator the part on the bottom cannot equal zero. If it does equal zero then it's undefined. So we know that X squared minus X minus cannot equal zero. Well what X values would make it equal zero. Um Well we don't know so we have to figure that out. So we'll figure out how to make it not equal zero by setting it equal to zero. Let's say x squared minus X minus 30 does equal zero. And then we'll know what X values. We'll make our function undefined so which one's it can't be? We can factor this so we know that's going to be X times X. And um recalling from previous lesson videos on how to factor these. We want to think about terms that multiply to equal negative 30 and add up to be negative one. And so we know that we could use negative six and plus five and we can just double check that really quick. X. Times X. The first terms X squared outside terms plus five X. The inside terms minus six X. And the last terms minus 30 combine like terms X squared minus X minus 30 equals zero. And yep that checks out. So we've got our two factors and we can set those individually equal to zero. We don't need a parentheses. So just X minus six equals zero and X plus five equals zero, solve for X. Real quick we get X equals six and X equals negative five. And those are the two values for X which will give us zero in the denominator. So X can be anything in the world except negative five and six. How do we write that in interval notation? Well we know that it can go from negative infinity all the way up to negative five But it can't be negative five. So we have a parenthesis there and that's in union with -5. All the way up to six but it can't be six and that's in union with six. All the way up to positive infinity. We look at our answer choices. That is answer choice A. Now if you are wondering why isn't it answer choice C. With this upside down you will we recall that the upside down U. Is the intersection of those values. And therefore that would mean where all those values come together and that's nothing. Those values have nothing in common. So it can't be C. Or D. That leaves us with that match of answer choice. A Well done. We'll catch you on the next one

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