In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the ...

Question

In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x)= = 9x/(x - 4), g(x) = 7/(x+8)

Accepted Answer

Hello. Today we are going to be using the two given functions to find F divided by G of X and identify its domain. The problem tells us that the function F of X is equal to 18 X divided by X minus six. We are also told that G of X is equal to 16 divided by X plus 10. How can we use these two functions to find the function F divided by G of X? Well, F divided by G of X is telling us that we need to take the F of X function and divide it by the G of X function. So let's go ahead and follow this process. We'll take the F of X function which is 18 X divided by X minus six and divide this function by G of X which is 16 divided by X plus 10. Currently, the function F divided by G of X is in the form of a complex fraction. In order to simplify the complex fraction. We'll want to turn the overall denominator to the value of one. In order to do this will multiply the denominator by its reciprocal which in this case will be X plus 10 divided by 16. Now keep in mind because we are multiplying the overall denominator by this reciprocal, we must also do the same to the numerator. So we will also multiply the numerator by X plus 10 divided by 16. In the denominator, we can cross reduce the terms. And that is going to allow us to have a denominator of one in the numerator. We will be left with 18 X multiplied by X plus 10, divided by 16, multiplied by X minus six. There is one thing that we can do to further reduce this function. Notice that 18 and 16 are multiples of two. So we can divide 18 and 16 by two. And that will allow us to reduce the function to be nine, X multiplied by X plus 10 divided by eight multiplied by X minus six. This is going to be the simplest form of F divided by G of X. Now, what we need to do is we need to identify the function's domain. Now recall that the original F of X and G of X functions are in the form of rational functions. So we're going to need to find the restrictions of these two functions and combine it with the restrictions of the current F of F divided by G of X function. So if you don't remember how to find the domain of a rational function, what we need to do is we need to take any value that's in the denominator, which for F of X is going to be X minus six set, it equal to zero and solve for X. That will give us the restrictions of the denominator that cause the denominator to go to zero. So let's go ahead and start by finding the domain of F of X. Now, F of X has the value of X minus six in its denominator. So we are going to take X minus six set it equal to zero and solve in order to solve for X, we'll add six to both sides of the equation. And that will give us X is equal to positive six. What this means is that the value of positive six causes us to get an undefined value for F of X. So the domain of F of X is going to be all the values from negative infinity to positive six union, positive six to positive infinity. Now, what we are going to do is find the domain of G of X. The function G of X has the values X plus 10 in its denominator. So we are going to take X plus 10, set it equal to zero and solve for X. We'll subtract 10 to both sides of this equation. And that will give us X is equal to negative 10. So negative 10 will cause us to get an undefined value for G of X. And the domain is going to be all the values from negative infinity to negative 10 union, negative 10 to positive infinity. Now let's go ahead and find the domain for F divided by G of X F divided by G of X has the value eight multiplied by X minus six in its denominator. So we'll take the denominator and set it equal to zero and solve for X. In order to eliminate this coefficient of eight, we are going to divide both sides of the equation by eight. That will allow us to have a coefficient of one on the left hand side and zero divided by eight will give us the value of zero. So we have X minus six equal to zero, we will add six to both sides of the equation and that will give us X is equal to six. And that tells us that the deno the domain of F divided by G of X will have the same domain as F of X. Now, what we wanna do is we wanna take the union of all of these domains. If you're not sure how to do this, it'll be helpful to draw a number line and label all the restrictions of X. So we know that positive six and negative 10 are our current restrictions. So we want all the values going from negative infinity, not including negative 10, going up to six, but not including positive six and going to positive infinity. And the way we label these sections is going to be all the values from negative infinity to negative 10. Then we want to include the region from negative 10 to positive six and include the region from positive six to positive infinity. So that means that the domain of F divided by G of X is going to be all the values of X from negative infinity to negative 10 union, negative 10 to positive six union, positive six to positive infinity. That's going to be the final domain for F divided by G of X which is nine X multiplied by X plus 10, divided by eight multiplied by X minus six. And that tells us that the solution to this problem is going to be B So I hope this video helps you in understanding how to approach this process. And I'll go ahead and see you all in the next video.

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In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x)= = 9x/(x - 4), g(x) = 7/(x+8)