In Exercises 31–50, find fg and determine the domain for each fun...

Question

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

Accepted Answer

So for this problem it says for the given functions F of x equals five X squared plus 33 X minus 14, MG X equals X plus seven. Find F times G and write the domain. So let's go ahead and find one. F. Times G is F F times G. Now, what we're gonna do first actually factor out our F of X. So let's go ahead and look at our fx. We have F X 25, X squared plus 33 X minus 14. Let's go ahead and try and factor this. We can use this called the A. C method, which helps us back your polynomial or A is our first term The coefficient. So that would be five C is our last term coefficient, which is I'm 14. So a times C is going to be equal to five times 14, which is 70. Now let's find the factors of 70. Well look here, we'll have 172 and five and 14 And seven and 10. Let's see if adding any of these two factors together will equal our middle middle number right here, 33. And we also have to remember one of our numbers is actually negative because of our negative 14 here. So let's go and take a look. 1 70. There's not really a way we can get to 33 for 1 70. Let's look at two and 35. There actually is a way we can get there from two and 35. So let's use those two numbers. Now we will rewrite our polynomial five X square. Now we want to find what numbers add up to equal 33. We have 35 and 2. 1 of them has to be negative. Let's say we have plus 35 x minus two X. And finally we'll just bring down our -14. Now we have that we can go ahead and factor this out even further. We look here for our first two terms. We have five X squared plus 35 X. Let's factor out things that are similar In both of those terms. We look here we can check out at five actually giving us X squared plus seven X. And then we look again we have an X. That's in common can take out the X five X times X plus seven. Now if you look at our next part -2 X -14 we can actually take out a negative two out of both of them. It's -2 times x plus seven. Let me get minus two X plus seven. Down here. Left with five X times X plus seven minus two times X plus seven. Now combine our outside terms and we can combine our inside terms as one term. You look at like this, you get five x minus two by itself and X plus seven by itself. So that is our factoring breakdown for F X. Now let's go ahead and multiply it by R G X. So f times G. That's equal to five X minus two times X plus seven. And finally our G. X. Expo seven. We will multiply by expo seven. This could be even further be written To just have FG equals five x - and X plus seven squared. We have two X plus sevens. So this is our equation. Now I want to check what the domain is now, domain looks like to see if we have any discount of new these or anything weird inside of our polynomial. If there's any restrictions at all. If we look here, we have a linear function times another linear function squared. In other words, a linear time is a quadratic. Since we're just multiplying those two functions together, we know this domain right here for five exponents to that domain is actually all real numbers or Nice and Sandy to infinity. The same thing for the extra seven and that's squared. There's no restriction. So this is also all real numbers. eight infinity to infinity meaning our answer. Our domain. This next infinity to infinity. Our equation Five X squared. I'm sorry. Five x minus two times X plus seven squared. Which is answer. B. Okay, I hope that will solve the problem. Thank you for watching. Goodbye

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