Hi, everyone. Welcome to our lecture on common functions. We're just going to list in this lecture some basic functions that you've probably seen before, but we're going to describe them in full detail. We're going to give their graphs. I'm just going to sketch the graphs on the lecture, but I encourage you to go over to one of the websites that we talked about and see the graphs in perhaps a better detail, more drawn to scale. The first functions that we talk about are linear functions. These are the pretty common ones that I think most of us have seen. These are of the form f of x equals mx plus b. We're going to have a lot more to say about this later. You might have known that m is the slope and b is the intercept. But this is the general form of a linear function. They're called linear function because their graphs are perhaps are surprisingly lines. The two that I want you to know for this one here is the very basic one when m is 1 and b is 0. So this is f of x equals x. This is a function whose graph you should know when you see the function. If you think about what this is, is the identity function, so you plug in 0, you get 0; you plug in 1, you get 1; negative 1, you get negative 1. It's a nice diagonal line that goes right through the origin. This is great. Let's say though, like a little more than just the graph itself in terms of what we can describe, what are some attributes of this graph, the domain of the graph. The domain, remember, is a set of all inputs. What numbers am I allowed to input into this function? What this function says, give me a number, or we turn it back. There's no problem with that at all. Its domain is all reals and its range, which we'll denote with a D and an R, its range is, you give me a number, I give it back, it's exactly the same as the domain. I'm writing reals with this double bar R. That's a common way to write all real numbers. Some folks like to write it using interval notation. That's perfectly fine as well. You can write negative infinity to positive infinity and then this is important. Make sure you put the parentheses on the infinity signs. Either way, you like to write it with the R with a double bar or the negative infinity to infinity is perfectly fine. This function f of x equals x has a graph for line. Its domain is all reals, its range is all reals as well. Other things you might want to know if you have a graph that you can talk about, like what are its x-intercepts, what are its y-intercepts? I'll review that with x-int, and then y-int as well. From the reminder, the x-intercept is where the graph crosses the x-axis. For this particular graph, it goes right through the origin, so we'll list the origin down here at 0, 0. The y-intercept is where it crosses the y-axis. This easy function here turns out to be exactly the same. Other things you might want to ask yourself are certainly what are the end behavior or what does this function want to do as x gets large? That's going to be a nice defining property of functions here. As x gets large, again, the function is just returning what the value input is. So we say as x goes to infinity, as x gets large, the function gets large as well. As x goes to infinity, the function goes to infinity, and same thing as x gets very small, the function gets very small as well. So we can write it out as a separate sentence if I can fit it here on the slide. As x goes to negative infinity, f of x goes to negative infinity. This is a nice little summary of the property of this function. It is related to one other function, so let's do f of x equals negative x. These are the only two linear functions that I want you to know everything about f of x equals negative x. What did I do? I changed the coefficient on the x, I changed the value of m. This is now a negatively sloped line, slope has a negative one. For example, if I give you x is 1, you return negative 1. If I give you x is negative 1, you negate that and you return positive 1. Same thing goes right through the origin, nice diagonal line, only this one is going from high to low as you move from negative to positive. It has very similar features, its domain again, all reals, and its range, since you're just negating every single number is all reals as well. Its x-intercept and its y-intercept, once again, are both the origin. Where does this graph cross the x-axis? Where does this graph cross the y-axis? So far everything looks the same. But what's different about this one? The end behavior. As x gets large, then the function, where's the function want to go down into the abyss. The function goes to negative infinity and as x gets small, so what's the end behavior now as I get negative, the function actually gets quite large. It blows up and goes to positive infinity. Positive infinity, the same thing you just want to stress that as x gets small, this gets large. There are many linear functions, but these are two that I want you to know everything about. The next common function you'll see is what's called quadratics. The general quadratic is a polynomial of the form ax squared plus bx plus c. You may know this from having to memorize at some point the quadratic formula, which solves for roots of this type of equation. Again, there are lots of variables, lots of numbers here, you can pick for a, b, and c. So they are infinitely many kinds of these, but I just want you to know the basic one, for the most part, this is f of x equals x squared. This is the parabola. This comes up often enough that we should just know everything about it. Let's collect all the facts right here, right now and you can keep these handy and use it as a reference sheet. We've drawn this as before. Y equals X squared is the parabola, it goes right through the origin, has a nice U shape to it. Let's list some things about it. What is its domain? What numbers am I allowed to square? Can I square positives? Sure, 2 squared is 4, 3 squared is 9, etc. Can I square 0? Sure 0 squared is 0. Can I square negative numbers? Sure, negative 3 times negative 3 is positive 9. You can square anything you want. Its domain is all reals. Its range. Now, this is interesting, what is going on with the range here? The range is the set of outputs. It's a set of y values. If you think about it on the y axis, what are the numbers you get? Can I ever get a negative number if I square something? Think about this for a minute. Can you ever get a negative number? The answer is no. If you research the real numbers, which is what we're going to do for our domain, I can certainly get 0 right across the origin there, so 0 squared is 0, so 0 is an output of mine and I get any large number I want, I can definitely get positive as I square things. This is important. I can get 0 when I get the number, when the number's included in my set, I put a bracket. I don't use open parentheses. When I can't get the number, like infinity is not really a number, its this concept, like you never quite get there, use open parentheses. So be very careful here with your brackets and parentheses. Bracket means include, parentheses means don't include the number or you can't get to the number. Infinity always gets a parenthesis because you can never quite get there. What else do we want to know? Let's talk about intercepts. What is the x-intercept and what is the y-intercept? Once again, this graph nice, little parabola goes right through the origin. It has one point where it crosses the x-axis, which is exactly the same point where it crosses the y-axis. Not every graph will have this, not every quadratic will have this, but this parabola does. What's the end behavior? As x goes to infinity, as x gets large, where's the function want to go? The function f of x goes to infinity. As x gets small, as x gets really small, where's the function want to go? If you're a little bug and you're walking left on this thing, on the parabola, where do you go? Off to infinity as well? Different range, same intercept, same domain, and different end behavior to help distinguish what it means to be the quadratic. Again, lots and lots of quadratics out there, but this is the one I want you to know, this will be the foundation for most of the other ones that follow. Next up is a cubic. F of x is a_x cubed plus b_x squared plus c_x plus d. A cubic is a way to say a degree three polynomial. Friendly reminder, the degree of a polynomial is the highest exponent that appears. So when the highest exponent was a one, we had a linear polynomial when the highest exponent was 2 we had a quadratic polynomial, and now the highest exponent is 3 we have a cubic polynomial. Once again, I just want you to note the simplest cubic that there is, and this is x cubed. When the coefficient's 1 and the rest are to 0, just go to old x cubed. This function, I like to call it the disco function, why? Move your hand in this motion as I draw it, stand up, put on your 70s outfit, you'll see why. Goes from low to high and like the other graphs that we've seen so far, crosses the origin right at 0. Think about this. This is like you give me a number and I will cube it. Zero cube 0 times 0 times 0, that's 0, 1 cubed 1 times 1 times 1, that's 1, 2, 2 times 2 times 2 is 8, etc. You can throw in positives, you can throw in zeros, you can throw in negative. Your domain on this guy is all reals, your range. Now, this is interesting, if I take a positive number and cube it, I get back a positive number. If I take 0 and cube it, I get back 0. If I take a negative number, three negatives make a negative. You actually get all real numbers back for your range. My set of inputs is all reals, and my set of outputs is all reals as well. What is the x and y-intercept? A little theme going on here for these introductory functions. It's the origin once again, so it goes right through 00. What's the end behavior? If you're a little bug, you're walking on the graph, where you're heading up, you're heading up, you're heading down and going somewhere else. We're going to infinity. As x goes to negative infinity, as x gets really small, three negatives make a negative. I also had to negative infinity. This is another graph that I want you to know, I want you to have a picture when someone's talking about a cubic behavior. This is the graphs that should come in your mind. Again, these properties and facts about it as well. Let's look at another common function. This is a little unlike some of the other ones, a new one square root here, so let's just write this guy out. This is f of x equals the square root of x. In this class it's going to make more sense for us to do a little algebraic switch. Friendly reminder, the square root of x is the same as x_0.5. You'll see it either way. If I draw the graph, the sketch of this graph as taking a square root of a number gives you back a positive number. The square root of 0 is 0. Some people get confused with that, like am I allowed to do that? Yes, you're allowed to do that. Can't really see that color, so let's mix that up a little bit. If I take 0, I get back 0. Then square root of 1 is 1, the square root of 4 is 2, and this graph just goes and rolls and off it goes. It grows slowly up like a slow steeping mountain. This is the graph y equals the square root of x. Let's give this thing some basic properties. It's a little more rounded than I drew it, but this is the idea. Now, what is its domain? What do you allow to plug into the square root function so that we get an answer back. We can only plug in in this class zero or positive. Plug in zero is fine, plug in positives, fine. You can't plug in negative numbers. Our university at this class studies real numbers. If you know a little bit about complex numbers or imaginary numbers, yes, they're thing. Yes, they happen when you throw negatives under a square root. But we're not going to talk about them in this class at least for a little while. When we talk about functions, the underlying assumption is that everything is a real number, so our domain is only going to be zero to infinity. This is our first graph where you have a restricted domain. This is not the entire real number. We can't throw a negative under a square root. Its range is also, give back a positive number if I take a square root and I also do get back zero if I plug in zero, the same zero to positive infinity. Brackets on the zero, parentheses on positive infinity. X-intercept and y-intercept. I promise this will always be the case. We go right through the origin again. For end behavior, as x goes to infinity, if you go a little back and you're walking on the graph. You're getting there slowly is not as fast as the other ones, but you certainly do climb and climb forever and ever and ever, then that means that the function, the square root function wants to go to infinity. We don't really talk about end behavior as x gets small because it's not in the domain. I won't even put what happens, the question doesn't make sense. If I said to you, what happens as x goes to negative infinity of this graph, because there is no graph back there. It's not included in the domain. Square root function, a little similar, a little different than the other ones, know this one as well. Here's another one, 1 over x. This is what's called a rational function. You got a ratio of terms, so they call these rational functions. This is the parent category that 1 over x falls under. Its graph is neat. It's a reciprocal or give me a number I give you back the reciprocal. If you give me one, I give you 1 over 1, that's 1. If you give me two, I give you a 1/2. I worked my way, as x gets larger and larger and larger, I gave you a 1/2, 1/3, 1/4. This function gets small, still stays positive. Positive 1/2, positive 1/3 taking reciprocals in numbers and get to 100, like 1 over 100 gets really small. If you start working your way closer to zero, like let's say 1/2 and take its reciprocal, like flip it, you get 2. If you take 1/3 and flip it you get three. As you get closer to the y-axis, the graph actually shoots up and you have an asymptote over here. The same thing on the other side. You get this very similar behavior, just everything's negative. If you take the reciprocal of a negative, you also get back a negative. This is a nice interesting function. It's the first one we see a little bit of an asymptotic behavior. The y-axis is certainly an asymptote as is the x-axis. You think lines the function gets close to but never quite gets there. This is a new little function is a good one to know. It's got some symmetry to it. It's cool. Let's think about what its domain is. What numbers am I allowed to take reciprocal of? Can I take positive like 3, 4, 5? Sure, 1/3, 1/4, 1/5. Can I take negatives, sure. I just can't take zero. You can't take zero. There's a couple ways to write this depending on how fancy you want to be. The simple way to do is to say x not equal 0. It's like saying the negative though it's what I can't do versus what I can do. If you want to get a little fancy, there's a couple of different ways to do this. If you know your interval notation, you can do like negative infinity to zero but don't include zero, and then I want zero to infinity again. I want to like this first interval, negative infinity to zero or zero to infinity and you can union like it that now we're getting fancy. You can union the two. This into union of two intervals is exactly the same as x not equal to 0, exactly the same. This is called interval notation. Another way to do it if you want set notation, now we're getting like super fancy here. You could write all real numbers except, and you could throw zero. It's a set though, so you have to use those curly brackets. This says the same thing. It says basically anything you want except zero. Any one of these three is fine, quite fancy. Take you position. That's the domain. The range, this is also anything, I'll say y not equal to 0. All your outputs are fine. You'll just never get zero. To start with zero, you'd have to have zero in the denominator and flip it and you can't divide by zero. For all the reasons why x is not zero and the domain is all the reasons why y is not zero in the range. X-intercepts and y-intercepts. Finally, not the origin. This graph you can see from my sketch does not go through the origin. What are my x-intercepts and what are my y-intercepts? To find this, ask yourself where does the graph go through the x-axis? You look for second, wait a minute, x-axis is a horizontal asymptote and never goes through. That's cool, so there's no x-intercepts. Y-intercepts, where does the graph cross the y-axis? It doesn't. You don't have any, none. That's new. Finally, not the origin. We have domain, we have range, we've intercepts, what's our end behavior? This one as x goes to infinity. Now, this is interesting. If you're a little bug, and you're walking where are you headed towards? The function wants to go to zero. Never quite gets there, will asymptote, but it's going closer and closer and closer. As x goes to negative infinity, the function also wants to go to zero, this time though from below. If you're a little bug and you're walking left, you're approaching the zero, or you're approaching y equals 0, the x-axis. Here's one also, a lot of new things finally happening here as our functions get a little more complicated, we have no intercepts and our functions don't want to go to infinity finally as x gets large. Here's absolute value. We've seen this before, I think, but let's just write it down. This is where you put bars around the number, and this means give me back the positive version of the number. For example, if I hand you the absolute value of five, well, that's just five. It's already positive. If hand you zero, well, that's also to zero. The thing is if it's negative, well, then you just give me back positive two. You don't return any negatives. If you start graphing this thing, you'll pick some points and start graphing. So 1 gives you 1, 2 gives you 2. You start getting the line y equals x. At least if you're positive, what exactly what this means, it says really do nothing in the bars, don't contribute anything and you get y equals x, you go right through the origin. But if you're negative, then I start getting positive values. Negative one returns positive one, negative two returns positive two, and you get a line, basically negate yourself, to negate a negative gives a positive, and you get the line y equals negative x. You get this V shape. It's two lines. Slope negative one and slope one, and it goes right through the origin. We're back to that case. It's a nice little function, has a letter V to it. The domain, all reals range zero to infinity, can certainly get back any of those things. Then x and y-intercept, back to our common case here. Where does the graph cross the x-axis, where does it across the y-axis? Right at the origin. As x gets really big, then the function wants to go to infinity, and as x gets really small, then the function also plug in a large negative number, get back a large positive number, wants to go to infinity. These are the big ones that we want you to know. You've probably seen these before. These are all the ones that we're going to assume that you have some working knowledge of. These will come up in conversation and your uses as a reference. One thing I want to mention, and this is just a super brief intro. We will spend a lot of time in this course on exponential logarithms, but you hear it all the time. People talk about like models, or growth, like this grows exponentially, the exponential function, the most common one. I just want you to know that graphs here. We don't need to go into all the details here. But the shape of the graph because it's going to come up, of course, e_x. Now, e of course is a number and it looks like a letter, but it's like 2.718 in chain. A little less than three. Exponential growth. When you see something like this, an exponential growth, it starts off low and goes high. Think about like compound interest or something else. But exponential growth, populations tend to exhibit, spread of diseases tend to exhibit exponential growth. You start off low and you go high and all these numbers are nasty numbers if you plug them into the calculator, there's only one nice little point and that's the y-intercept, and that's if you plug in a zero. A number raised to the zero, you're going to get one. You can go through what we'll talk about, the domain and the intercepts and everything else and why it has an asymptote back here later. But I just want you to know the general shape of this graph. Again, go have fun with this thing. Go graph it, play around that [inaudible] if you want e_x. Then just as a general shape, I want you to know what the logarithm looks like. Someone's talking about it at this point, do you have an idea? The most common logarithm, again, we'll see the other ones is the natural log of x, and that graph is the exponential graph but flipped, when you turn your head a little bit and flip it over the line y equals x. This is also lots of nasty decimals that you get by plugging into your calculator. However, what you can do is, there's only one really nice point and it's the x intercept this time. That's at one. If you plug in one, so the natural log of 1 will give you 0. This is your x-intercept. On the exponential graph, if you plug in e_0, you get 1, and that's your y-intercept. It's just general shape of graph. I just wanted to do some lots lots more to say about exponents and logs, but we'll get there eventually. But this is just a good general place to start and introduce these two functions. Great job on this one. See you next time.
