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) = 2 + 1/x, g(x) = 1/x

Accepted Answer

Hello. Everyone in this video we're going to be looking at this practice problem where we want to find the composite function F divided by G. And write the domain. And were given that F of X is equal to 12 plus three divided by two X. And G. Of X. Is equal to three divided by two X. So in this case our first step is going to be to try and find the composite function F divided by G. So F divided by G is the same thing as the F function F. Of X, divided by the G function G. Of X. And were given the values of F. Of X. And G. Of X. So if I plug those in my numerator, F of X becomes 12 plus three divided by two X. And my denominator G of X becomes three Divided by two x. So in this case you can see that I have fractions in both my numerator and my denominator. So I'm going to rewrite this in a linear division to make it easier to simplify. So if I do that I have my numerator 12 plus three divided by two X. And that's being divided by my denominator which is three divided by two X. So now I have my division written out in a linear format and from here a hack to remember is that you can change a division between fractions into a multiplication if we switch the numerator and the denominator of the second fraction. So now I'll have 12 plus three divided by two X times two X divided by three. So no it is how I switched the numerator and the denominator between the fraction. So now I can multiply the fraction instead of having to worry about dividing it and now that I have this multiplication I just need to make sure to multiply this fraction to each of these components. So first component I have is 12 times two X. Divided by three. So I'll go ahead and write that. And then I also have a plus three divided by two x Times two x divided by three. So if I simplify this You can see that 12 times two is 24. So I'll have 24 x divided by three. And then my second fraction you can see that I have a two X. In the denominator and the numerator of the second fraction. So that becomes one. And then I also have a three in the numerator and denominator of the fraction. So that becomes one. So this simplifies two plus one. And if we look at this first part, 24 X divided by 3, 24 divided by three is eight. So if I completely simplify this, I have eight X plus one. So f divided by G. Is actually just eight X plus one. So this becomes my composite function. And now I want to worry about finding the domain. So recall that the domain is the range of X values. That will result in a real number. So whatever number I plug into this X I will get a real number back because there's no possibility of me plugging in a number and getting something that's undefined. So I can say that the domain for this particular function would be negative infinity to positive infinity. However, because it's a composite function, I have to make sure that any excess that were not part of the domain in the previous functions are also not part of the domain in this new composite function. Because the composite function is built on those older functions. So we have to look at F of X which is 12 plus three divided by two X and G. Of X. Which is three divided by two X. And because the domain only cares about the X values, we want to look at the parts of the functions that have X values. So in the case of F of X we only care about this fraction three divided by two X and G of X. Also the same fraction three divided by two X. So really we're only looking at the three divided by two X. And when you think of three divided by two X, you see that X is in the denominator. And if the denominator of a fraction is equal to zero will get an undefined answer. So we want to make sure that X is never zero to make sure that the denominator is never zero. So we can say two X cannot equal zero because we don't want the denominator to equal zero. And if we solve for X, we get X cannot equal zero. So now we have a value that we need to exclude from the domain because we know that X can equal any value besides zero. So I now know that X can go from negative infinity to zero. And I'll do a parentheses because like we said, X cannot equal zero, but it can get really close to zero and then I'll do a union because X can still go from zero to positive infinity. So now this becomes my true domain for the function that we built. So this becomes my final answer for the domain for the composite function. And my composite function f divided by G. Is eight X plus one. And you can see that that aligns with answer choice D. 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) = 2 + 1/x, g(x) = 1/x

Key Concepts

Function Division

Domain of a Function

Rational Functions

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