### Video Transcript

Simplify the function ๐ of ๐ฅ equals the fraction ๐ฅ minus seven over ๐ฅ squared minus three ๐ฅ minus 28 minus the fraction ๐ฅ minus seven divided by seven minus ๐ฅ, and determine its domain.

After we copy down the function, the first thing that weโre going to wanna do is see if we can factor this denominator, ๐ฅ squared minus three ๐ฅ minus 28. Letโs do that! Can we factor ๐ฅ squared minus three ๐ฅ minus 28? Since the leading coefficient is one, we know that itโll be ๐ฅ as the first term of both factors. Now I need the two factors of 28 that multiply together to be negative 28, and those same factors need to add together to equal negative three.

First step, one and 28. Then we have two and 14. Three isnโt a factor. From there, weโll take four and seven. I recognize that the difference between seven and four is three. If I make the seven negative and the four positive and add them together, Iโll get negative three. And if we multiply them together, four and negative seven, weโll get negative 28. These will be the second terms in our two factors here.

We take those two factors and we put them in the denominator to simplify the function. From here, letโs stop and talk about determining the domain. The domain is the set of all ๐ฅ-values for which output a real ๐ฆ-value. When weโre working with fractions that have variables in the denominator as functions, we need to be careful to remember that our denominator cannot be equal to zero.

This means that ๐ฅ plus four cannot be equal to zero; ๐ฅ minus seven cannot be equal to zero; and seven minus ๐ฅ cannot be equal to zero. What we want to do to determine the domain is to find the places that would make ๐ฅ plus four equals zero and find the places that ๐ฅ minus seven would be equal to zero. We need to solve for ๐ฅ here. To do that I can subtract four from both sides of the equation, and then I determine that ๐ฅ cannot be equal to negative four.

To solve for ๐ฅ minus seven, we add seven to both sides, and we say that ๐ฅ cannot be equal to seven. Letโs go ahead and check one last place, seven minus ๐ฅ. We can start by subtracting seven from both sides. We see that negative ๐ฅ cannot equal negative seven, and then we multiply both sides of the equation by negative one, and we see that ๐ฅ cannot equal seven. What we can say about our domain is that itโs true for all real numbers with the exception of negative four and seven.

So we write all reals minus the set of negative four and seven. Okay, letโs get back to simplifying this function. What we can see in one of our terms is that we have ๐ฅ minus seven in the numerator and ๐ฅ minus seven in the denominator. These values will cancel each other out; ๐ฅ minus seven over ๐ฅ minus seven equals one. This will leave us with one over ๐ฅ plus four. Now I look at this second term, and I wonder what to do. ๐ฅ minus seven over seven minus ๐ฅ. Can we simplify this?

With the help of this subtraction symbol, we can simplify ๐ฅ minus seven over seven minus ๐ฅ. Hereโs what weโll do. Instead of saying minus here, weโll say plus negative one times ๐ฅ minus seven. What you should recognize here is that we have not changed the value of this problem at all. Weโve simply change the format. Instead of using subtraction, weโll use addition, and weโll multiply by negative one.

Iโll multiply ๐ฅ by negative one, which gives me negative ๐ฅ. Then Iโll multiply negative one by negative seven, which gives m positive seven. Bring down the denominator. If we look closely at negative ๐ฅ plus seven, we see that we can actually change the order here. Instead of saying negative ๐ฅ plus seven, we can say seven minus ๐ฅ over seven minus ๐ฅ. Okay, bring down the one over ๐ฅ plus four.

And if we clean this problem up and just copy down seven minus ๐ฅ over seven minus ๐ฅ, do you see that that equals one? Now weโre trying to add one over ๐ฅ plus four plus one. But to add a fraction, we need to have a common denominator. So Iโm going to write one as ๐ฅ plus four over ๐ฅ plus four. Itโs still equal to one, but now we can add these two pieces together. When weโre adding fractions with a common denominator, we add the numerator, one plus ๐ฅ plus four, and the denominator will stay the same, ๐ฅ plus four.

We can add the one and the four together. One plus four equals five, and our denominator stays the same, ๐ฅ plus four. The function ๐ of ๐ฅ could be simplified to ๐ฅ plus five over ๐ฅ plus four, and the domain here is all real numbers minus the set of ๐ฅ equal to negative four and seven.