In Exercises 82–84, find f + g, f - g, fg, and f/g. f(x) = x^2 + ...

Question

In Exercises 82–84, find f + g, f - g, fg, and f/g. f(x) = x^2 + x + 1, g(x) = x^2 -1

Accepted Answer

Welcome back. I'm so glad you're here. We've got two functions. We've got f of X equals x squared minus x minus four. And we've got G of X equals two, X squared plus five. Those are two functions. And what we're going to do is find the sum, the difference, the product and the quotient of those two functions. Let's start off with the sum. So we'll write our F of X term, X squared minus x minus four and that's going to be added to our G of X term, which is two X squared plus five. And now we just combine like terms we've got an X squared plus two X squared is going to give us three X squared. This is the only X term. So we've got minus X and then minus four plus five is going to give us plus one. We've got a lot of answer choices here. So to keep track of everything, let's eliminate answer choices that don't match. So for the some three X squared minus X plus one will eliminate answer choice is B and D. Great. Now let's find the difference. We are going to take x squared minus x minus four in this time it's minus and we'll put a parentheses two x squared plus five. So that we remember to distribute this negative sign to both terms in the second function. So we've got X squared minus x minus four minus two X squared and minus five. Now we can combine like terms. So we've got the X squared minus two, X squared, which will give us negative X squared. Just bring that in that minus X again and minus four minus five this time which will be negative nine. Let's check and eliminate some of the answers that don't match up. So again we can eliminate B and D. Okay that didn't really help us out getting the answer too much. But we know we're on the right track with either A or C. Now let's find the product. So we've got X squared minus x minus four times R. G of X which is two X squared plus five. And we can distribute this X squared times the two X squared will give us two X to the fourth X squared times the five will be plus five X squared negative X times the two X squared will be minus two X. Cute negative X times the five will be minus five X negative four times the two x squared will be minus eight X squared and negative four times the five will be minus 20 combining like terms this is the only X to the fourth. So two X to the fourth. We've got a minus two X. Cute. We have two X squared terms. So plus five minus eight will be minus three X squared, Bring down this minus five X and bring down our minus 20. Well that does match our A and B choices. However it is equivalent to R C and D. Choices. So we haven't eliminated anything with that one. Now let's look at the quotient the division of these two terms. So we take our F of x, r x squared minus x minus four, divided by r G of x, which is two X squared plus five. Can we simplify anything? They're fact or anything? We can't That's as simplified as that one is going to get, so that eliminates our C and R. D choices, which leaves us just with our answer choice. A well done, we'll catch on the next one.

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