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 divided by G. And write the domain and were given the F of X is equal to 27 X divided by X plus nine and G of X is equal to nine divided by x minus seven. So in this case F divided by G is the same thing as F of X divided by G of X. And in this case F of X and G of X are both fractions. So I'm going to write it in the more linear format of F of X divided by G of X. So if I do that, F of X is equal to 27 X divided by X plus nine Divided by G of X, which is nine divided by X -7. So recall that in this linear format I can change this division to multiplication by switching the numerator and denominator of the second fraction. So instead of being nine divided by x minus seven, it's gonna be X minus seven divided by nine. But now it becomes easier to combine my functions because it's a multiplication. So I have 27 X times X minus seven, Divided by nine times x plus nine. And from here you can see that the 27 can be simplified to three and this 9-1. So it becomes three X times X minus seven divided by X plus nine. So F divided by G is equal to this equation. But now that I have this equation, I want to find the domain. So in this case the domain is the range of X values. That allows this function to be give me real answers. So because we have a fraction, I want to make sure that the denominator doesn't equal zero. So X plus nine cannot equal zero. And if I solve this you can see that X cannot equal negative nine. So I have a limitation there. The other thing I need to check is that because this is a composite function that is built off of F. And G. I need to incorporate their domain into my domain. So I already know that X cannot equal negative nine. And if I look at F of X, you can see that I have the same equation where X plus nine cannot equal zero. So X cannot equal negative nine, which I've already taken into account. But G of X has a different equation where nine divided by X -7 and now X -7 cannot equal zero. So if I solve for X I get that X cannot equal positive seven. So because my composite function F divided by G is based off of the F function. And the G function. My ex can also cannot equal seven as well. So now I have two limitations or values that my X cannot equal and need to be excluded from the domain. So for my domain X can go from negative infinity to negative nine Again with parentheses because I cannot equal negative nine exactly. But I then can go from -9 to my next limitation, which is seven. And again, parentheses because I cannot equal seven. Exactly. And my final union will be from seven to positive infinity because after seven there is no limitation. So this becomes my final composite function with my domain. And you can see that this combined function and domain align with answer choice B. So hopefully this video helped and see you next time.

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