In Exercises 31–50, find fg and determine the domain for each fun...

Question

In Exercises 31–50, find fg 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 times G. And we want to write the domain of that composite function. 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 order to get this problem started, we're going to need to find the composite function first F times G. Because we can't write the domain if we don't know what the function is. So we're going to start by finding F times G and F times G is the same thing as the F function F of X times the G function G of X. And were given the values of both F of X and G of X. So I can go ahead and plug those in. So F of X is 12 plus three divided by two X. And I need to multiply F of X times G. Of X and G of X is three divided by two X. And from here you can see that the vex has two components, the 12 and the three divided by two X. So I need to make sure that I multiply G of X three divided by two X to each of those components. So f times G is equal to Times three divided by two x. And then I have to do the second component of the F of X. So I have plus three divided by two x. Times three divided by two x. And if I carry out this multiplication I have 12 times three which is 36 divided by two x plus three times 3 is nine divided by two X times two X. Is for X squared. So now you can see I have two fractions and they have different denominators. And in order to add them I want to have the same denominator so I need to find the least common denominator. And from here you can see that four X squared is the same as two X squared or two X times two X. Which means that this first fraction is actually only missing a two X. So in order to get the same denominator in both of these fractions I'm going to multiply this first fraction by two X over two X. And my second fraction stays the same nine divided by four X squared. So if I do this multiplication I have 36 times two x which is 72 x Divided by two x times two x. Which is for ex squirt. So now I have 72 X divided by four X squared plus nine divided by four X squared. And now I have the same denominator so I can go ahead and combine my enumerators to leave me with x plus nine Divided by four x squared. And if we wanted to simplify even farther you can see that from the numerator I can take out a nine. So if I do that I'll put nine on the outside of my parentheses and 72 divided by nine is 8. So I have eight X plus nine divided by nine is one. So I have nine times parentheses eight X plus one and parentheses divided by four X squared. So that's my new composite function of times G. And from here I need to find the domain. So recall that the domain is the range of X values. That gives me a real answer. So I want to find all the values of X. That give me a real answer in this equation. So the first step is going to be looking out where X is present. So we have X. In the numerator. And in the denominator. And if we plug in negative values into this ex that's in the numerator we'll just get negative values back or positive numbers if it's less than this positive one. So really the X in the numerator won't cause any undefined answers. However if we get a zero in the denominator will get an undefined answer. So we need to try to avoid that. So we want to make sure that this denominator does not equal zero. So we can write an expression four x word cannot equal zero. And from this expression we can solve for X to give us the range of values that we want to avoid. So if I solve for X I'll divide by four. So X squared cannot equal zero. If I take the square root of both sides, I get X cannot equal zero. So this is saying that if I plug in zero into this expression, I'll get an undefined answer. So my domain cannot include zero. So if I write that in interval notation, that means X can go from negative infinity 20 and I'll do parentheses because I can't do a bracket since X can't equal exactly zero, but I can do a union since it can go from zero to positive infinity. So that becomes my domain. And normally this is all you would have to do for a domain. However, because this is a composite function, we have to make sure that this domain also is valid in the previous functions F of X and G of X. So if we look at F of X and G of X, just going to write them here, F of X is equal to 12 plus three divided by two, X And X is equal to three Divided by two x. So in this case, recall that the domain only cares about the X value. So I'm only going to look at the components of the functions where X is present and in both of X and G of X, the X is present in this fraction three divided by two x. So really we have the same scenario where X is in the denominator, which means that we want to X two not equal zero. And if we solve for X, we got X cannot equal zero, which is the same condition that we already solved for. Which means that we don't need to consider any other conditions for our domain. So this becomes our final domain, negative infinity to zero, union with zero to positive infinity, and f times G R composite function is equal to nine times parentheses eight X plus one and parentheses divided by four X squared. And if we look at the answer choices, 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 fg and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x

Key Concepts

Function Composition

Domain of a Function

Rational Functions

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