Find f/g and determine the domain for each function. f(x)= = 8...

Question

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 that F of X is equal to X divided by X plus nine and G of X is equal to nine divided by x minus seven. So in this case the first step is going to be to find F divided by G and F divided by G is the same thing as F of X divided by G of X. And were given the values of F of X and G of X. So I can plug those in. So 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. And I can rewrite this division in linear form. So I have 27 X divided by X plus nine divided by nine Divided by X -7. So recall that when you have two fractions, one in the numerator and one in the denominator, we can actually rewrite this division as a multiplication. If we switch the numerator and the denominator in the second fraction. So our new fraction becomes X -7 Divided by nine. So that now we have 27 x times X minus seven divided by X plus nine times nine. So if we do that multiplication We have 27 x times X -7 and nine times x plus nine From here, you can see that this nine and this 27 can be simplified to be three since 27 divided by nine is three. So we're left with three X times x minus seven divided by X plus nine. So now we've simplified f divided by G as much as we can. So that becomes F divided by G. And now we want to try to find the domain and recall that the domain is the range of X values. That will ensure that this equation remains real. So In this case any value we plug into this X will allow this function to be real because even if we plug negative 100 we still get negative 300. Or even if we plug Positive 100 we get positive 300 and the same case with this numerator. The only issue we would have is in the denominator because if we have a denominator that is equal to zero. Then we get an unreal answer. So we want to make sure that our denominator is never equal to zero and we can do that by saying X plus nine cannot equal zero. And from there we can solve for X. So I'll subtract nine from both sides. Leaving X cannot equal -9. And now we need to write this in interval notation. So from negative infinity two negative nine X can exist. And then again from -9 to positive infinity X can exist. So in this range, notice how I used parentheses to indicate that I can't equal negative nine. Exactly. But I can get really close to negative nine as long as X is not equal to negative nine. So this interval becomes my final domain and it ensures that when whatever value I plug into X will not give me an invalid answer because my denominator will not be equal to zero. So here you can see we have F divided by G. And our domain. So these become our final answer. Where F divided by G is equal to three X times X minus seven, divided by X. Plus nine. And our domain is negative infinity to negative nine. Union with negative nine to positive infinity. And you can see that that aligned with answer choice D. So hopefully this video hoped and see you next time.

Find f/g and determine the domain for each function. f(x)= = 8x/(x - 2), g(x) = 6/(x+3)

Key Concepts

Function Division

Domain of a Function

Finding Restrictions

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