Hi, again. I want to talk today about functions and the domain of functions. So we have this domain, or definition of domain, is the set of all real numbers where f of x is also a real number. So the domain is a description, right in here, real numbers x where f of x is also a real number. So in other words, it's a list of all inputs where the outputs are real. Real numbers. Now in some ways it's a list of all inputs where the outputs might be in some ways nice or not super complicated. So that's the idea. And we want to get an idea of where this domain is for some functions. So when we talk about finding the domain, in my world or in my description, I usually begin with all real numbers. So in my mind, I'm beginning with a real number line. And for functions, you should start with this real number line. And the domain is everything. OK. And then I start removing stuff. Remove stuff that cause 0 in the denominator. So any time you have an x in a denominator, that's a possibility. And you have to look for that. So this one, I'm looking for denominators with x in them or with a variable. OK. The next thing that we look for, at this point, or another function that we know, remove any values of x that cause a negative under an even root. So I'm looking for, here, even roots with variables. OK. And then the fourth item is to consider, especially if you're talking about word problems or application problems, consider the context that you're talking about and exclude any values that don't make sense within that context. A lot of times, for example, if you're talking about just a geometry problem, you're trying to find the length of the sides of a rectangle. The sides of a rectangle can't be negative, so you would exclude those. The final thing is domain is about input values for sure. It says something about the output values. We know that all the output values are then real for these input values, but it's a description of input values. And we use interval notation to write out the solution for domain. So let's look at a couple examples for domain. So I want to do find the domain of f of x is equal to x squared plus 5 divided by x plus 2. OK? Now we have these three things that I'm checking for. I start with all real numbers. And then I want to check for these three things. The first thing I check for is, are there any denominators where x could cause a 0? And the answer to that question is yes. There are some [? denominators. ?] In fact, I have this x plus 2 in the denominator of this function. Now I don't want that to be equal to 0, so I'm going to find the places where it is equal to 0. And that is x is equal to negative 2. So when I talk about the domain of this function then, I have this entire real line. Sorry, the negative 2 is a little bit too far. And that I have to remove. And I use, from growing up, remove that from the domain by an open circle. And then I continue to go down the list. For example, are there any square roots? In this case, there are no square roots in my function. And it's not an application. So really, I just have this one point that needs to be removed. And the rest of this is the domain of this function. The domain in interval notation should go from minus infinity up to negative 2 and-- I'm going to skip negative 2-- negative 2 up to infinity. You might also see this written with a union sign-- it's the set notation, if you took 106-- like that. So either one of those would be a solution for the domain of this function. Let's do another problem. Find the domain of f of t is equal to t plus 1 over-- sorry, I don't want that parentheses there-- t squared minus t minus 2. OK? Again, I go through my list. I identify that I have denominators that could cause a 0. Yes. But no square roots. And it's not an application. So really, I'm just looking at this denominator issue. And I take the denominator and set it equal to 0 to see where it is 0 and will cause a problem. This is a quadratic function. I need to solve it, this equation, quadratic equation, in order to find it. I think this is factorable. T minus 2, I think. And t plus 1. Does that work? I think that works. And then, I split using the zero-product property. t minus 2 equals 0. And t plus 1 equals 0. And I get t is 2 and t is negative 1. So I want to go ahead then and remove those. And I'll just draw a real number line. And then we'll put it into interval notation. So here, maybe in red, I have to remove negative 1, and I have to remove 2. Otherwise, everything on the real line is in the domain. It should look like this in interval notation, minus infinity to negative 1, negative 1 to 2, and 2 to infinity. Open parentheses all around. And you're welcome to use the word and instead of this union sign if you would like. Let's do one more involving the square root sign. Oops. Find the domain of f of x is equal to the square root of 2x plus 6. OK? So I don't want 2x plus 6 to be less than 0. That's what I don't want to have happen. That causes a negative under this square root. So I'm, again, going through my list. There's no denominators here. There is a square root. And this is not an application problem. So I don't want that to be negative. And I solve this inequality. There we go. x is less than negative 3. This is what I don't want to have happen. So that means x less than negative 3 should be excluded from the domain and taken out. So otherwise, 3 can be in there. And the rest of this can be in there. So for those of you who don't like thinking in this kind of reverse way on these problems, you can also solve this inequality here. 2x plus 6. And I want 2x plus 6 to be greater than or equal to 0. And you get something like that. They will give you the same and lead you to the same place. In interval notation, this should be closed bracket negative 3 to infinity for the domain. OK. That's it for domain. We'll talk more about domain in the context of application problems in that video. And we'll hit domain again when we get to log functions. Those have restrictions as well. Please let me know if you have any questions.