In Exercises 31–50, find f−g and determine the domain for each fu...

Question

In Exercises 31–50, find f−g and determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1

Accepted Answer

this problem says for the given functions F x equals five X squared plus 33 X minus 14 and G X equals X plus seven. Find f minus G and write the domain. So let's find an f minus G. S. So we have f minus G. R f equation is five X squared plus 33 X minus 14. And r g equation is X plus seven. So I will subtract in parentheses X plus seven. I want the parentheses there. So we make sure the negative goes to both terms of G of X. Now let's go ahead and simplify this. You have five X squared which says the same Plus 33 x -14. This should be our negative negative X and -7. Now we simplify even further. We have our five X squared which just comes down, we have a 33 X minus and single X. That is plus 32 X now and negative 14 minus seven which is gonna give us negative 21. So this is our equation but we can actually factor this out even further. So let's look at this, we have five X squared plus 32 X minus 21. We use the A C method where a Is the number on our leading coefficient. In other words, our highest exponents which is squared. That is going to be 5C. Is the number of our final term which in this case is 21. We're gonna find a times c. That is 21 times five. That is a 105. Now I want to find what two numbers can multiply the equal 105. So let's run through them. We have 105. of course we don't have a two but we do have a three. So if we did three You would get 35 3 and 35 And we have five and 21. So I actually stopped there because I see which pair of we want to use. If you look here we're gonna find two numbers here that are going to add up to equal 32. Now keep in mind one of these numbers will be negative because we have a -21. So let's look at these here I'm gonna say we have three and 35 equaling a 32 Because we can have a positive 35 and a negative three. So let's go ahead and rewrite this. So we have five X squared. Now we do a plus 35 X minus three X minus 21 for that. 35. And the three give us the 32. Now we want to try and factor out some things from our terms. If you look here we have five X squared plus 35 X. We're looking at those first two terms. You see there's some things in common. There's a five in both of those numbers. If I would take five take five out of there I could X squared plus seven X. Where we take a five out to 35 and then we look again we have an X. We can take out from both. We have five X times X plus seven. Now let's check our other other terms over here. If you look here we gotta take out a negative three out of both terms because three can come out of 21 ST negative three and X plus seven negative three X plus seven. Now we just combine all of a sudden on the outside and all of a sudden it's on the inside of the parentheses. So write this again are five XR -3 will come together And our Expo seven will come together. Our equation It's five X -3 and X 47. Now we want to check with the domain is we want to see if there's any restrictions on either of these terms. We look here we have five x minus three and X plus seven. Those are both linear terms. And there's no restrictions really on these functions. So if you look at that you can say that it's gonna be negative infinity to infinity because neither term has a restriction being our domain is all real numbers or not infinity to infinity. And our equation of five X minus three times X plus seven. Making this answer answer. A Okay I hope that you solve the problem. Thank you for watching. Goodbye

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