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)= = 8x/(x - 2), g(x) = 6/(x+3)

Accepted Answer

Hello, everyone in this video, we're going to be looking at this practice problem where we want to find F minus G and right P domain and were given that F of X is equal to 18 X divided by X plus nine and G of X is equal to nine divided by x minus seven. So in this case our first step is going to be to find f minus G. So recall that F is equal to the F function. So F of X minus the G function which is G of X. So we know the value of F of X and G of X. So we can actually plug that in. So F of X is equal to 18 X divided by X plus nine and that's being subtracted by G of X, which is equal to nine divided by X -7. So now we found what F minus G. S and we want to keep it this way to keep it in the simplest terms. And now we want to try to find the domain of this function. So in order to determine the domain, recall that the domain is the range of X values that will result in a real answer. So we want to make sure that whatever value of X we plug in will give us a real answer. And because we're working with fractions here, you can see that the only place where we wouldn't have a real answer is if the denominator, whether it be X plus nine or x minus seven or zero. Because if we have a denominator of zero, we have an answer that's not defined. So we're trying to avoid that situation. So we can set up two equations where we want X plus nine to not equal zero and we want x minus seven to also not equal zero. So if we solve that, I'll subtract nine from this first one, I get X cannot equal negative nine. And if I add seven to my second one, I have X cannot equal positive seven. So in this case you can see two values that are being excluded from the domain. So how do we write that? An interval notation? So if we look at the number line And say this is -9 And this is seven, I'll draw a circle on negative nine and seven because I know that X cannot equal these values, but it can go to negative infinity And it can go between -9 and seven and then it could also go to positive infinity. So looking at this number line, you see it's easier to visualize the intervals. So if we convert this to intervals we see that X can go from negative infinity to negative nine. And it can also go I'll draw union because it can also go between negative nine and seven And then there's this last interval. So I'll draw another union where it can go from seven to positive infinity and you can see that I've drawn parentheses where there's the negative nine or the seven because I know that X cannot equal negative nine or seven. So this becomes my final range of X values that will result in a real number or my domain. And we've already solved for f minus G, F minus G is equal to 18 X, divided by X plus nine and -9, divided by X -7. So this becomes my final answer. And you can see that this aligns with answer choice B. So hopefully this video helped and see you next time.

In Exercises 31–50, find f−g and determine the domain for each function. f(x)= = 8x/(x - 2), g(x) = 6/(x+3)

Key Concepts

Function Subtraction

Domain of a Function

Combining Domains

Watch next