 Welcome back to our lecture series Math 1050, College Algebra for Students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Misseldine. In this first video for lecture three, we're going to talk some more about functions, right? But with a lot of the basics of functions now out of the way, we're going to start getting to some more of the nitty-gritty algebra side of the things. I hate to use such a negative term here because algebra itself is a pretty impressive tool to help us solve many mathematical and even non-mathematical problems. But in the previous two lectures as we were in section 1.1 about fundamentals of functions, we were focusing on well, the fundamentals of functions and we avoided the use of any real algebra so that we really can focus on the function fundamentals themselves and not get distracted by the algebraic part of things, which is important but a separate part. Algebraic representation is only one way to represent a function. Now we're going to break that convention, and we're going to start using these algebraic functions. When we talk about an algebraic function, what we mean is it'll be a function described algebraically, typically of the form f of x equals some type of algebraic expression right there. Evaluation of the function will be, well, your algebraic expression will have some variable in it probably like an x usually, and you'll replace that x value with the number hand, and that's how we evaluate functions algebraically. So let's consider the quadratic equation, the quadratic function f of x equals two x squared minus three x. Function evaluation, like if we want to do f of three here, what this means is everywhere in the equation for the formula that you see an x, you replace that with a three. So you get two times three squared minus three times three. Each of those x's was turned into a three, and then we didn't follow the arithmetic calculations here, three squared is a nine, three times three is likewise a nine, two times nine is 18, 18 take away nine is therefore nine. And so then this algebraic statement tells us that the function evaluated at three gives us a nine. So this function will send the number three to the number nine, and that's how it does it, and it does it using the formula provided. If we were to do f of one, it's the same basic idea, we'll take two times one squared minus three times one. One squared is one, times that by two is two, three times one is three, you get two minus three, which is negative one, and that's then the evaluation of the function. That's all there is really to it. Now, one nice thing about these algebraic functions is that we can evaluate the function at specific numbers in its domain, like three and one, but we can also evaluate these functions at other expressions themselves, like we can look at f of negative x. What does that mean? f of negative x would mean you're gonna take the function f of x from above and everywhere you see an x, you're gonna instead substitute in a negative x. So we get two times negative x squared minus three times negative x, and so uniformly everywhere you see an x, you're gonna replace it with a negative x. I would recommend putting parenthesis around the negative x so that we get the correct order of operations. For example, when it comes to the negative x squared here, the negative x squared means negative x times negative x. That'll be a double negative, and hence it'll actually turn into a positive. We end up with a two x squared, and likewise we get this negative three and negative x, that's again another double negative. So we get two x squared plus three x, and this is the function evaluated negative x. Now you might wonder, why would we evaluate the function negative x? I thought we put numbers inside the machine and it spits out a number, right? Well, there are gonna be reasons why one would want to plug in an algebraic expression. For example, this is a little bit of foreshadowing here, but if one wanted to reflect, say, the function across the y-axis, replacing x with negative x does exactly that, and we'll talk about that at some future date here. If we evaluate the function at h, f of h here, that just means replace each x with an h. You get two h squared, replace the h for the x, two h squared minus three h, and that's all there is to it. If you evaluate and never put a box around the other answer, if you evaluate the function at some algebraic expression, the output will be an algebraic expression, for which case we can do something like that. All we did was just change the variable name, but there'll be times where we wanna change the variable name. We might wanna change from x to h, and reasons like that will become more clear much, much later on. At this moment, we're playing the karate kid for which Mr. Miyagi is telling us to wax the car, wax on, wax off, and out of context, it gets a little bit confusing while we're doing it, but all of these have justifications, and at the moment we're just practicing sort of our muscle memory, but not necessarily with our physical muscles, but with our brain here, just practicing our algebraic skills. There will be times where when we have an algebraic formula like f of x, that we want to do algebraic operations onto it, but not necessarily evaluating the function at some value x, but what if we actually want to operate on the formula itself? Three times f of x means we take three times the formula two x squared minus three x, in which case I would recommend at this moment we distribute the three onto both pieces, in which case we then get six x squared minus nine x, so we can multiply it by three, right? What does it mean to do negative f of x? That just means times negative one times two x squared minus three x. Again, you would distribute the negative sign and get negative two x squared plus three x, switch to sign there. So we can start doing operations on the outside of the function. We can also do some on the inside of the function, like let's go back and let's replace x with three x. What that means is we'll take two, instead of an x we'll replace it with a three x, put parentheses around it to make sure we get the scope correct, and then minus three times three x, and then we simplify this thing algebraically. Three x squared means three x times three x. We get two times nine x squared, and we'll get minus nine x, and so times the two by the nine, and we end up with 18 x squared minus nine x. And so we get all of these evaluations in the following manner. These are things we've probably done in other algebra classes, so we're finally able to start doing them now, these algebraic functions. All right, here's another example. What if we wanna substitute in for x, x plus three, some binomial? Well, it's the same basic idea. We're gonna replace x with x plus three, make sure you put parentheses around the x plus three, it's very imperative you do it this time. You'll get negative three times, instead of x we get an x plus three. That's what it means to evaluate at x plus three. And then the only thing I would recommend here is maybe simplify it, I guess. I mean, it's good in the way it is, but if we were trying to combine some like terms and such, you're gonna have to foil out the x plus three. And so by the usual foil method, what we mean here is x plus three squared is x plus three times x plus three. To square something just means to multiply it by itself. And if you go through all the possible products, first x squared, outside three x, inside three x plus the last terms nine, you do all the possible foils like this. Combining like terms we would end up with, well, two times, we still have the two from before, two times x squared plus six x plus nine. And then we're also gonna distribute the three right here. So we end up with a negative three x minus nine. Make sure the negative sign comes with the negative three when you distribute it. So you get a negative three x and a negative nine. Then I would distribute the two onto this part right here, in which case we get two x squared plus 12x plus 18. And then we have the minus three x minus nine. So we're going to combine some like terms. It's like we are on Facebook, we're gonna hit that like button. 12x minus three x, we're also gonna take 18 minus nine. And so in the end, this function when evaluated at x plus three gives us two x squared plus nine x, 12 minus three. And then 18 minus nine is likewise nine. And so f of x plus three can simplify to be the quadratic expression. Two x squared plus nine x plus nine. And so it might not exactly be obvious why, but when it comes to algebraic functions, we can evaluate them at numbers, but we can also evaluate them at various different algebraic expressions. So these will all become clear later on why in the world would we wanna do something like this? We're gonna see in the not too distant future that f of negative x is exactly what we wanna do if we wanna reflect across the y-axis. Negative f of x will do a reflection across the x-axis. f of three x will do some type of horizontal compression by a factor of three. And then three f of x will be some type of vertical stretch by a factor of three. f of x plus three is gonna be if we wanna shift the graph to the left by three, we're gonna learn all about these. And this just gives you some of the reasons one would do this, there are others than this, just to kind of give you some idea. At the moment, wax on, wax off Danielson. We're gonna practice these skills and apply them in the not too distant future.