For the pair of functions defined, find (ƒ/g)(x). Give the domain of each. See Example 2. ƒ(x)=2x^2-3x, g(x)=x^2-x+3

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Hello everyone. We're going to find the function F divided by G of X. Given the functions F of X equals four, X squared minus X and G of X equals X squared minus five X plus seven. And then we're gonna determine what the domain is. So first thing I'm gonna do is I know this means I want F of X over G of X. I'm going to write those expressions in so four X squared minus X over X squared minus five X plus seven. Looking at it, I recall that I should try to reduce this but there does not appear to be anything that will reduce. So this is the first part of our solution. Four X squared minus X over X squared minus five X plus seven. Now I need to find the domain. So the domain of F of X when I look at it, there is no square root. There's no denominator. So the domain of F of X is all real numbers. So then looking at G of X again with G of X being that it has no square root or radical of any kind and no denominator, it will also be all real numbers. However, looking at our solution to F of X over G of X. Obviously we have some variables in the denominator there. So we are going to set that denominator equal to zero to see if there's any real number of values that will not work out. So we are going to do zero equals X squared minus five X plus seven. And as much as I wish this was factory ble it's not So we're going to use the quadratic formula. Recall that in the quadratic formula we work with zero equals a X squared plus Bx plus C. So a in this case is gonna be one B. is gonna be negative five C. Is gonna be seven. And that the full format is X equals negative B plus or minus the square root of B squared minus four A. C. All over two. A. Alright so let's find out if X has a real number so X equals negative B. So the opposite of negative five is positive five plus or minus the square root of B squared. So negative five squared minus four times A. Which is one times C. Which is seven All over two times a. Which is one. Alright so I'm gonna clean up what's under my radical. So I get five plus or minus. The square root -5 Squared is -4 times one is 4 times seven is over to. Alright so continuing cleaning up under that radical, I have five plus or minus the square root of negative 3/ And I'm gonna factor the square root of negative one out of my radical. So x equals five plus or minus I radical three over to. So when I look at that I have an eye in there, I means it's an imaginary number. Our domain can only have real numbers. So this means that there is no real number. They'll make this undefined because the only number that would make it undefined is imaginary. So the domain of F. Of G. F over G. Of X. It's gonna be the union of our F. Of X. Domain and our G. Of X. Domain, both of which were all real numbers. So this remains all real numbers. So our domain is all real numbers. And this is answer choice. B have a good day.

For the pair of functions defined, find (ƒ/g)(x). Give the domain of each. See Example 2. ƒ(x)=2x^2-3x, g(x)=x^2-x+3

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Key Concepts

Function Division

Domain of a Function

Polynomial Functions