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) = 3 − x², g(x) = x² + 2x − 18

Accepted Answer

for the given functions F x equals 12 minus three X squared and G X equals three X squared minus 18 X plus 27. Found left by the budgie and rock domain. Let's go ahead and write left by the budgie G equals r f 12 minus three X squared divided by three X squared minus 18 X plus 27. They want to simplify this division will do that. We're gonna factor up both of our polynomial. We look at the top we have 12 -3 x squared. I can say we actually can take out a three out of both terms. I took up three out, we look at 12 12 or 34 taking a three of the three X squared gives us negative squared. This is divided by the bottom. Now the bottom of one, a factor of three, X squared minus 18 X plus 27. So the first thing we should know is that look at the factors of three, those are just one and three. We're gonna have three X and X. Now, next we can look at the sign of our last number, the sign of our last number is positive. However, the sign of the middle number is negative. That implies that we're gonna have two minuses in the middle because if we were to re foil this and we multiply our factors, we should have two negatives multiplied and equal a positive. So now let's look at the factors of 27, we have one and nine And I'm sorry, not one in 9, one in 27 and nine and 3. So you figure out where our nine and three will go, let's say the nine goes here and the three goes there but we foil that. We should get the same thing back out three x times X which is three X squared three x times negative three which is negative nine X negative nine times X. Which is negative nine X. And negative 93 which is positive 27. Let's get those three X squared minus 18 X plus 27. So those are the credits factors. We write it at the bottom three, That's a three X -9 x minus three. So now we have this we can actually factor out even further. So can actually factor out our four minus X squared and r three x minus nine. So you go over here you have four minus X squared by the difference of two squares formula. This states a squared minus b squares. Has the factors A minus B A plus B. So if we say our a squared in this case it's four. four square the forest is too. And say to S R A. Now RB is a squared of X squared, that would just be X two minus X. Since we have difference of squares. This should also be two plus X. We can simplify this again by looking at our two minus X. We're gonna rewrite both of these with X in front we get negative X plus two times X plus two, we have a negative in here let's factor out that negative better than negative from this X plus two. We get negative x minus two Times X-plus two. Let's go back to relax minus nine. We have three x minus nine. We can factor out a three from both terms. We get three times x minus three. We write this again, we get three times X -2. With a negative negative will now go up front. X plus two, divided by three times X minus three Times X -3. We can do a little bit more simplification by crossing out R three's first. So this three and this three then we get negative on top x minus two. X plus two. On the bottom we notice we have two X -3 is now we can rewrite this as X -3 squared. So this is our simplified equation for F divided by G. I want to find what the domain is, domain. We want to check to see if our denominator is equal to zero and what X value makes it equal to zero. So let's go take the denominator of dysfunction and set it equal to zero. We have x minus three squared equals zero. Let's go and solve for X. You'll take the square root. Yeah X minus three equals zero and X. By adding three to both sides, you get X equals three. So the Mexico three will make the domain equal to zero, which means our function cannot be equal to three. So if we want our function to be not equal to three, that is every number except three, which is from native infinity, two, three and three to infinity. So just start coming. Our function is native X -2 times x plus two Over X -3 Squared. Made the answer to this problem. His answer. D. Okay, I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 31–50, find f/g and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 18

Key Concepts

Function Division

Domain of a Function

Factoring Quadratic Functions

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