Welcome back. We're going to start our session today on exponential functions. I'm sure you're a little familiar with these, but it's a good review and make sure we all understand everything we need to know about them. So let's remind ourselves what it means to be an exponential function. So when a function is exponential we'll say, if it is of the form with a number to the base f of x is exponential. Let's write this down. If it is of the form x equals a_x, and here's the thing, a is a fixed real number that's greater than 0. So again, here's a new example. Let's talk about something that is, and something that isn't. So how about 2_x? That is a nice exponential function. We'd like this function. The confusion that some students often get when talking about exponential functions, so let's give an example. Something that's not. If I gave to you f of x equals x squared, this is no, this is a power function. The behavior of these functions, I know they have a two and I know they have an x but it's the location that matters. In an exponential function, the number is the base, the variable is upstairs. In a power function, the variable is the base and the number is upstairs. Just to show you their graphs and how they look and feel differently. This is the graph y equals 2_x vs the graph of y equals x squared, which is the parabola. They have some similarities, but there's hopefully you see far from the picture that they are different enough. With any exponential function, of course, let's look at the domain. The domain is all reals. With my parabola, the domain is also all reals. Here's one of the differences. The range is going to be, now there's an asymptote back here, I never quite touch zero. It's going to be open interval zero to infinity. So positive values for this function. Over here, the range is going to be, and remember I can get zero. So it's actually zero to infinity. So this is where that bracket versus the parentheses makes a very large difference. There's an asymptote for y equals 2_x. It has a horizontal asymptote. We abbreviate this with the HA, and this is at the line y equals 0, and there's no asymptotes for the other one. So this is the two sources of confusion. Just keep in mind which is which. The exponential function has the number as the base. Now, when talking about exponential functions, there are three kinds. Let's talk about them here. So the first kind is if 0 is less than a is less than 1. So remember, the function y equals a_x is our exponential function and a is always positive. We don't talk about functions with a negative base that gets us into complex numbers. Remember, we don't talk about that in next class. So think of a is here's a half or a fourth or something like that. So if that's the case, the function is like one-half and it's being multiplied by a bunch of times. But the nice thing about this thing as you plug in zero whenever you're function is, just as an example over here, how about y equals one-half_x. So if you plug in zero, what's the intercept on the y-axis? You always get one. In this case here, the function goes from high to low. We say this is exponential decay. It's a decreasing function. It has intercept, so y-intercepts right at 0, 1. It has a nice horizontal asymptote also at y equals 0. Something of the x-axis is a dashed line. If the value of a is one, so in this case over here, we have the graph of y equals 1_x, this is the world's most boring function. If I take a number and one to that number. So one squared, one cubed, 1_4, you always get back 1. There's actually just a constant function that goes through here. This function is not really exponential, so technically it does satisfy the exponential property, but we don't think about it in that case, so we just tend to ignore it. It's just a little sad face over here. We don't study that example. The third case is if a is greater than 1. This is the one we like. This is the one that comes up bunch. So in this case here, see example like we had 2_x before, this one goes from low to high. It's an increasing function, has domain all reals, and has a nice y-intercept at 0, 1. Once again, it has a horizontal asymptote as we saw before. So these are the functions that come up. They come all the time in situations that model exponential growth, exponential decay, compound interest. We'll see a bunch of these examples in the homework and problems. But hopefully, you've seen some of these before. When you have these down, we can start asking you to manipulate and play with the graphs to draw these things. So this is the graph of exponential functions, we should know how they behave, we should know what they do, let's look a little more complicated example. So f of x equals minus two_minus x minus one. This is an exponential function, it's got that two in the base and the variables upstairs, it's got some pieces to it, so let's play around with this thing, let's look at the graph, let's look at the domain, let's look at the range. So how do I look at this thing and understand how it's working without rushing to my calculator? Well, first and foremost, what is the parent function? It's y equals 2x. It's built from that. In our head, we should have the graph of y equals to x. Goes right through the y-axis at one, has a nice horizontal asymptote, a zero, low to high. So from here it goes y equals 2_minus x. So I've replaced x with a minus x. Think about what that does for a second. If you take a function and you put in a minus sign, this is a reflection about the y-axis. This is why if you have even symmetry, like y equals x squared, if you replace it with a y with x_negative x and there's no change to the graph. This is a reflection about the y-axis. So this graph becomes, so remember, here's the y-axis on top, to go everything on the right becomes everything on the left, goes like this. You can also think about this as 1 over 2_x or one-half_x. So this is exponential decaying, goes from high to low and this is what's building up at that. There's another piece to this function. There's a negative in front. So what happens to a function if you just throw a negative in front? This is another reflection, but not about the y-axis. This actually is a reflection about the x-axis. It takes whatever's up and shifts it down or flex it down. So the little part above the x-axis becomes the little part below and this graph goes below and crosses instead of at one where it crossed before, now crosses at negative one. Almost done. What happens when I do y equals minus two_minus x minus one? What is a minus one do to the graph? This shifts the graph, this is a shift. So it shifts up, down, left or right, again, think of some other examples you may know, this takes the whole graph and shifts it down one. So the dash line, the asymptote that was at zero, now becomes a nice dashed line at minus one, and the asymptote that was at zero gets pulled down and this thing is still faced down, but instead of crossing at minus one it crosses at minus two. So this is the final picture, and when you look at this thing, I want you to see the stages, I want you to think, what happens to this thing as it grows? The reason why we study exponential functions, the reason why they come up so often is because there's a very, very famous and popular exponential function with a very famous base and it is the number e, and then we will study this a little later too. I know it's weird to say this, but the number e, isn't e a letter? Yeah, but not for our cases. It's arguably the most important function in calculus, is f of x equals e_x. We will see this over and over and over. For those who have seen a little calculus before, you may know this has a very special property that's not shared with other exponential functions. So where does e come from? Well, it comes from a guy who studied, of course, it is the mathematician Leonard Euler, he was Swiss. So e for Euler, this is not Eular, put on your best Swiss German, this is pronounced like Euler. So Euler e, it's a number, what is this number? So he discovered it. It's a little odd if you think about it, to cover a number named after you but if you find a number with a very nice property, you too can have a number named after you. So this number e is 2.718, da-da-da, goes on forever. I'm originally from New York, so for me, 718, this is the area code of Queens, so for me it's two in the area code of Queens. It's two and change a little less than three, call it what you want. It is though, however, whatever you want to call it, it is bigger than one. So when we graph this function, we should know it behaves like any other exponential function with base greater than one. It goes from low to high, crosses right at one, because e to the zero is one and behaves like any other exponential growth function. Has domain all reals, has range, y is strictly greater than zero. So it's just like any other function, it just comes up so often due to its special property, which we'll study later, that we're going to call it out here and play around with it. All right. So let's do one of many examples to come with es. So I will give you a function, I want you to tell me the domain. You can pause the video and play around with it as you work it out. So here's a function f of x equals 1 minus e_x over 1 minus e_1 minus x squared. Slightly more complicated, then get all y equals e_x, but it has es in it, get comfortable working with this thing, and I would like to know, find the domain. What is the domain? So pause the video if you want and try to do it, let's see what happens. Here we go. Ready? When I look at this function, although I see lots of e's and x's, it's a fraction, it's got a numerator, and it's got a denominator. Everything I see in the numerator or denominator, I don't have concerns regarding the domain. I can take any number and square it, so that domain is fine. I can take any e to a number, a domain with all reals, I really have to worry about the denominator not equal zero. So denominator is not allowed to equal zero. Because if I get that, I'm dividing by zero, I'm breaking every rule in math. Nobody's happy with that. So I got to find the values of x where 1 minus e_1 minus x squared is equal to 0. Basically solve for x. When I find these points, then I will throw them away, and then I'll have all the good points left. So now, it becomes a good old solve for x and this should feel like a little precalculus problem. We have to solve for x with an e in the base, so let's move things over, and we get e_1 minus x squared equals 1. So how do I solve for e? Let's take ln of both sides, it's a good reminder of what the natural log does. Hopefully, you've seen this before. Pen is acting funny, sorry. So let me write this over, ln e_1 minus x squared is equal to ln of 1. Logarithms, friendly reminder, what is the natural log of one? That's zero. Talk more about that in a minute. The reason why you take ln in either both sides is because they cancel each other out, and the exponent falls down, and you get good old 1 minus x squared is equal to 0. Still not done, but hang in there. So move the x squared over, you get x squared equals 1. Now, be very careful here. A lot of students make this mistake. If you take a square root, you have to put plus or minus. Most people forget that and they just kick back the answer of one, but really the answer is plus or minus one. So there's two values here that are not allowed, plus one and minus one. So we can write that a bunch of ways, but what's a nice, easy way to do this? We can take all real numbers and throw away plus or minus one. That's the set way to do it. That's fancy. You can also say from minus infinity to minus 1 with a parenthesis union minus 1 to 1 union 1 to infinity. So I unioned and I used open parentheses, that means don't include one or minus one. Either one of these two work, this is set notation on the top and then interval notation on the bottom using union. So pick the favorite one you want. Let's do another example with exponential functions. Let's get very comfortable with these. I will give you a function. Let's start with, I don't know how about f of x equals 5_x? Switch it up. Let's find the difference quotient. Remember this? Find the difference quotient, f of x plus h minus f of x all over h. This is an expression that will come up later, but for right now, she's the complicated expression, manipulate the function, do something to it. Again, pause the video if you're working on this at home, try to plug it in, and then check with the answer. Here we go. This says take your function and where we see an x, plug in x plus h. So this becomes 5_x plus h. Take out the x, plug in x plus h. You don't need parentheses here, but usually, it's not too bad. Then minus the function, so minus 5_x over h. What's h? I don't know, some variable. Why is it there? Because it's there, I don't know. Is there anything I can do with this? Does anything cancel? Doesn't look like it. There isn't too much in all honesty that you can do. There is one thing that you can do, and I'm just going to do it to practice. If I have a number raised to x plus h, you can break that off as 5_x times 5_h. This comes from the rules of algebra. If you have the same base multiplied together, you add the exponent, so I'm doing it in reverse here. Everything else stays the same. The reason why I do this, because we'll use this form later, there's a 5_x in both pieces. So if I factor that out, I get 5_h minus 1 over h. Nothing cancels, but I was able to factor out a 5x. That may or may not have been immediately obvious from the start of this thing. It is what it is, what it is, remember, if I start off with letters in the question, I'm going to get letters back in my answer done here. We'll work with this later. But for now, this is the difference quotient with this one. Let's do one more example, and then we'll stop here, and let you guys go do some on your own. So true or false. I'm going to give you a function, it's different than the one we saw, although similar, this function 1 minus e_1 over x, 1 plus e_1 over x is an odd function. Now a lot of people get this wrong, mostly because they forget what it means to be an odd function. Friendly reminder, an odd function is if you have origin symmetry. That doesn't help because I bet you don't know what the graph of this function looks like so we need the algebraic property. This says if you plug in a negative, then the negative pops out. Think of it like x cubed, then the negative pops out. So that's really the question you're getting asked. You cannot look at this and just know the answer, this is a calculation question. So our goal is going to be to plug in f of minus x and check is it equal to minus the original function. I doubt many of us can do that in our heads, we have to work this out. So you shouldn't look at this and to say, "It's obviously true or obviously false," if you want to sound fancy and to hedge your bets 50/50 shot. But let's see what happens if we work this out. Here we go. Everywhere that I see an x, I'm going to plug in a minus x, and that's to all spots, numerator and denominator. What's going on with that? How do I clean this up? E to a negative exponent, you say, is there anything you can do? When you have a negative exponent, I remember the rule, if I have like a_minus n, it becomes 1 over a_n. Negative exponents become positive in the denominator, so let's do that. Let's write this as 1 over e_1 over x, and then 1 over e_1 over x, ignore that. So negative exponents become fractions. Now I took a complicated expression and made it into a complicated fraction, I have a fraction over a fraction, but that's okay, it's going to clean up. You say, "Well, is there anything I can do with this? Is there anything more to do?" Well, you have fraction. You can think of it like 1 over 1 fraction minus a fraction. What do you normally do when you have fraction minus a fraction? We simplify the fraction. So we have e_1 over x, e_1 over x. Let's write 1 with a common denominator, 1 over e_1 over x. That's my big fancy numerator, all over e_1 over x, e_1 over x. So that's just a fancy way to write the number 1 plus 1 over e_1 over x. So at this point in all, honestly, I still don't know where this is going. I don't know if it's true or false, but I know there's still algebra to do and I'm hoping that something cancels. There's enough e_1 over xs in here that I'd like it to be true or at least I'd like to see something happened to tell me if it's true or false either way. Anyway, so let's subtract, and add fractions. How does that work? Well, I have common denominators, so I add the numerators and you get e_1 over x minus 1 over e_1 over x. Then that becomes e_1 over x plus 1 over e_1 over x. So fraction divided by fraction. What do you do now? Well, this is a crazy exercise in algebra. You can keep change flip. You can multiply the top and the bottom by e_1 over x. Let's do that. So keep change flip. We keep the numerator, e_1 over x minus 1 e_1 over x. Division becomes multiplication and the denominator flips either 1 over x and you get e_1 over x plus 1. Now all of a sudden, because I'm multiplying es_1 over x actually do cancel. The last thing I'm going to do, because I want to know if this is negative f of x, I'm going to factor out a minus sign. So I'm going to get 1 minus e_1 over x, you could distribute the minus sign back in. If you want to see it, you get exactly either 1 over x minus 1 and the denominator doesn't change. I'm just going to flip the order so it looks like the original function I started with. So now do you see it? So if you factor out the minus sign, you actually do get back the negative of the original function. This crazy row through algebra somehow works out someway somehow. You can graph this too if you want to see the symmetry on it, but if you don't have access to a calculator on tests or something like that, you have to work this out this way. So the answer at the end of the day is this thing an odd function, this is true. This is a tough one just because the algebra involved, go over this example and the other ones before you start tackling the ones from the book or from the homework and just get comfortable working with exponential functions. All right. Great job. We'll see you next time.