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)= = (3x+1)/(x² - 25), g(x) = (2x -4)/(x² - 25)

Accepted Answer

Hello, today we are going to be using the functions F and G of X to find the function F divided by G and identify its domain. So let's go ahead and take a look at the given functions. We are told that F of X is equal to five X plus two divided by X squared minus 36. We are also told that G of X is equal to six, X minus 12 divided by X squared minus 36. We want to use these two functions to find F divided by G of X. So how can we begin this process? Well, F divided by G of X is telling us that we need to take the function F of X and divided by the function G of X. Let's go ahead and plug in F of X and G of X into this division. So we will have F of X which is going to be five X plus two divided by X squared minus 36. And that is going to be divided by the overall function G of X which is six X minus 12 divided by X squared minus 36. What this gives us is a function in the form of a complex fraction. In order to simplify this, we'll need to simplify the complex fraction. If you don't remember how to simplify a complex fraction, recall that we want to turn the overall denominator into one. In order to do this, we can multiply the denominator by its reciprocal, the reciprocal of the denominator in this case is going to be X squared minus 36 divided by six X minus 12. Keep in mind that we are multiplying the overall denominator by this reciprocal. And if we multiply the denominator by the reciprocal, then we must also multiply the overall numerator by the reciprocal. So we will also multiply the numerator by X squared minus 36 divided by six X minus 12. So how do we simplify this? Well, in the denominator, we can cross eliminate each of the terms. And that is going to allow us to have a denominator of just one in the numerator, we can also cross eliminate the terms X squared minus 36 and reduce those to be just one that is going to leave us with five X plus two multiplied by one in the numerator and one multiplied by six X minus 12 in the denominator. And that will allow us to simplify the complex fraction to just be five X plus two divided by six X minus 12. There is no further way to simplify this function. So that tells us that the final form of F divided by G of X is going to equal to five X plus two, divided by six X minus 12. Now that we have the function, we need to find its domain. Now recall that the domains or the original functions F of X and G of X are both rational functions. Furthermore, when we divided the functions together, we also got a rational function. What this means is that we need to find the restrictions of the domains of all three functions and the union is going to be the overall domain of F divided by G of X. So in order to start finding the, the domain, let's go ahead and find the domain of F of X and G of X first. So recall that F FX was given to us as five X plus two divided by X squared minus 36. And recall that the only value within a rational function that is not allowed to be in the denominator is zero. The reason for this is that any time we divide by zero, we get an undefined value. So in order to find the restrictions of the domain of F of X, we are going to take the values in the denominator which in this case is X squared minus 36 and set it equal to zero. After solving for X, this is going to tell us the restrictions of the domain on F of X So in order to solve for X, we are going to add 36 to both sides of this equation that is going to give us X squared is equal to 36. Next, we want to eliminate the exponent of two. In order to do this, we'll have to take the square root of X squared. And since we are taking the square root of X squared on the left side, we must do the same on the right side, that is going to allow us to simplify the left hand side of the equation to just be X. And on the right hand side, the square root of 36 will give us the value of six. But we need to be careful here. Remember that when you take a square root of a number, the output is going to be a positive and negative version of the perfect square. So that means that the square root of 36 is going to simplify to be positive and negative six. So this tells us that the values of positive and negative six are not allowed to be in the domain of F of X. And that means that the domain of F of X is going to be all the numbers going to negative infinity leading up to negative six, not including negative six union, all the values from negative six to positive six, not including positive six union, all the numbers from positive six to positive infinity. This is the domain for F of X. Now recall that G of X is given to us as six X minus 12 divided by X squared minus 36. Notice that the do the denominator of G of X is exactly the same as F of X. So that means that G of X is going to share the same domain as F of X. So now that we have the domains of F of X and G of X, we are going to find the domain for F divided by G of X. In order to do this, we'll take the values in the denominator of F divided by G of X and repeat the process of solving for X. That is going to be six, X minus 12 is equal to zero. In order to solve for X, we're going to add both sides 12. So that is going to give us six, X is equal to 12. Finally, we'll divide both sides of this equation by six and that is going to give us X is equal to two. So that means that two is the restriction on the domain of F divided by G of X. And that tells us that the restriction or the overall domain of F divided by G is going to be all the numbers from negative infinity leading up to positive two, not including positive two union, all the numbers from positive two going to positive infinity. Now what we want to do is we want to take the union of all of these denominators or all of these domains. And that is going to be the final domain for F divided by G of X. If we place this on the number line, the only restrictions are going to be positive and negative six and two. So on a number line, we want all the numbers from negative infinity going to negative six, going to positive two, going to positive six. And finally, leading to positive infinity. Furthermore, we cannot include the values of positive six, positive two and negative six in the domain of F divided by G. And that tells us that the overall domain of F divided by G of X is going to be all the values from negative infinity to negative six union, negative six to positive two union, positive two to positive six union, all the values from positive six to positive infinity quite the lengthy domain. But this is going to be the final domain of F divided by G. And recall that we solved F divided by G to be five X plus two divided by six X minus 12. With that being said the solution to this problem is going to be a. 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)= = (3x+1)/(x² - 25), g(x) = (2x -4)/(x² - 25)