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) = 3 − x², g(x) = x² + 2x − 17

Accepted Answer

this problem says for the given functions F x equals 12 minus three X squared. And G X equals three X squared minus 18 X plus 27. Find F times G and domain. So before I actually do this, I wanna factor out my fx and vfx. It'll make it a little bit easier to work with. So we have F X equals 12 minus three, X squared G x equals three X squared minus 18 X plus 27. Let's factor out our FX 1st. Now if you notice here, A three can actually fit into both the 12 and the negative three X squared. That's the only thing we can take out. Let's take out A three. Take it 234 minus X squared. We can factor this even further because we have four minus X squared. This is a difference of squares problem. So difference of squares that I live here, Difference of squares states that we have a squared minus B squared. The factors are equal to a plus B times a minus B. Well look here are a squared is four, R b squared is x squared. So our A plus B. R A is in the square of the four which is two. And our B squared is going to be X squared. So we have two plus X times two minus X. I'm going to write this again just to have the access in front. So we get x plus two times negative X plus two. In fact even farther by taking out the negative from the negative X plus two, which we get negative X plus two Times X -2. So this goes to negative three Times X-plus two x minus two. Now let's check our G. X. You notice here we have a polynomial three X squared minus 18 X plus 27. We can factor this by first taking our three X and r X first because of what the factors of three X squared. The only factors of three are one and three. When we get three X&X. Now look at our science. So we have a negative in front of the 18 x and a positive in front of the 27. So we need two numbers that multiply the equally positive 27 but two numbers will also add to equal negative 18. We can say this is going to be two minuses because the minus sense analysis will give us a positive and adding together two negatives will give us a negative. Now what you look at our 27, I'll write it down here as the factors one in 27 And nine and 3. So I think we want to use nine and three. Here look here we will get let's see three x minus blank X minus blank. Let's see the nine goes here and the three goes here, how to re foil this. You have three x times X which is three x squared three x times negative three which is negative nine x negative nine times X is also negative nine X. And finally negative nine times negative three X 5 to 27 which gives us three X squared minus 18 X plus 27. So those factors do work Go even further here we have a 3X -9. This can be further factored By taking out three. We have three. Now X take it at nine and three, it's three from nine gives us three again, x minus three. Finally we have two x minus threes multiply those together we get three X -3 Squared. So everybody here r f times G is equal to negative three times X plus two, X minus two times three times x -3 squared. We simplify this. We should have a negative three times a negative three which is a negative nine. And then the rest of the factors just follow each other again exposed to x minus two X -3 Squared. So this is our equation for f of X. I want to find out what the domain is. If you look here, we don't have any domain restrictions on any of our factors X plus two, x minus two, X minus three squared. Do not have any restrictions. So because of that our domain is from negative infinity 2050 that is our domain. And that is our equation. The answer to our problem is answers. See, okay, solve the problem. Thank you for watching. Goodbye

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In Exercises 31–50, find fg and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 17

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