Okay, welcome back. We're going to start our section 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. Okay, 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 the number to the base f(x) is exponential. Let's write this down. If it is other form x equals a to the 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 to the x? That is a nice exponential function. We like this function and the confusion that some students often get when talking exponential functions, so let's give an example of something that's not. If I gave to you f(x) equals x squared, this is no. This is a power function. Power function and behavior of these functions I know they have a 2 and I know they have an x but it's the location, location, 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 sort of differently. This is the graph y equals 2 to the x verse the graph of y equals x squared which is the parabola. And they have some similarities but there's hopefully you see 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 there's a tail back here. I never quite touch zero, so 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 0, so it's actually 0 to infinity. So this is where that bracket versus the parentheses makes a very, very large difference. There's an asymptote for y equals 2 to the x, it has a horizontal asymptote. We reviewed this with that HA and this is at the line y equals 0 and there's no asymptotes for the other one. All right, so this is the two source 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 and 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 to the 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're going to talk about that next class, so think of a as here's a half or a fourth or something like that. So if that's the case, the function is like one half and is being multiplied by a bunch of times. But the nice thing about this thing is you plug in zero whenever your function is just as an example over here. How about why equals 1/2 to the 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. Decreasing function, it has an intercept so y intercept, right at (0,1) and it has a nice asymptote, horizontal asymptote, also add y equals 0. So think of the x axis is a dashed line. If the value of a is 1, so in this case over here we have the graph of y equals 1 to the x. This is the world's most boring function. If I take a number and one to that number, so one squared, one cubed, one to the fourth, you always get back one. This is 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. So just a little sad face over here. I'm not really, we don't study that example. The other third cases, 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, this example like we had 2 to the x before, this one goes from low to high. It's an increasing function, has domain all reals. It has a nice y intercept at (0,1). Once again has a horizontal asymptote as we saw before. So these are the functions that come up, they come up all the time in situations 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. And when you have these down, we can start asking you to manipulate and play with the graphs to draw these things. So this is a graph of exponential functions, we should know how they behave, we should know what they do. Let's look at a little more complicated example, sp f(x) =- 2 = -2 to the -x -1. This is an exponential function, it's got that two in the base in the variables upstairs, it's got some pieces to it, so let's play around with this thing look at the graph, let's look at the domain, let's look at the range. So how do I look at this thing understand how it's working without rushing to my calculator? Well, first and foremost, what is the parent function? It's y = 2x, it's kind of built from that in our head we should have the graph of y = 2x, goes right through the y axis at 1 has a nice horizontal asymptotes at 0, load high. Okay so from here it goes y = 2 to the -x, so I've replaced x with a -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, 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 to negative x, there's no change the graph. This is a reflection about the y axis, so this graph becomes, so where is the y axis of 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 to the x or one-half to the x. So this is an exponential decay goes from high to low, and this is what's building up with that. There's another piece of this function, there's a negative in front, so what happens to a function if you just throw in 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 is up, and shifts it down or flex it down. So the little part above the x axis becomes a little part below, and this graph goes below, and crosses instead of at 1 where it crossed before now it crosses at -1. Almost done, what happens when I do y = -2 to the -x -1, what is a -1 do to a graph? This shifts the graph this is a shift, so you shift up, down, left to right again think of some other examples you may know, this takes the whole graph and shifts it down 1. So the dash line the asymptote that was at 0, now becomes a nice dashed line at -1. And the asymptote that was at 0 gets pulled down, and this thing still face down, but instead of crossing at -1 it crosses at -2. 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 okay? And 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 we will study this a little later too. And I know it's weird to say this, but the number e say is an e a letter? Yeah, but not for our cases, so it is arguably the most important function in calculus, is f(x) = e to the x. You will see this over and over and over, and for those who've seen a little calculus before, you may know this has a very, very special property, that's not shared with other exponential functions. So where does e come from? Well comes from the guy who studied, of course, is the mathematician Leonard Euler, he was Swiss, so e for Euler. This is not Euler, 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 have 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 dot dot dot dot dot goes on forever. I'm really 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 3, call it what you want. It is though however, whatever you want to call it is bigger than 1. So when we graph this function, you should know it behaves like any other exponential function with base bigger than 1. It goes from low to high, crosses right at 1, right? Because either the zero is 1 and behaves like any other exponential growth function. Has domain all reals, has range y strictly greater than 0. 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 example with e's. One of many examples to come with e's, so I will give you a function. I want you tell me the domain, and 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- e to the x, over 1- e to the 1- x squared. Slightly more complicated and got a y equals e to the x, but it has e's in it, get comfortable working with this thing and I would like to know, find the domain. What is the domain? Okay, 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 about regarding the domain. I can take any number and square it, so that domain is fine. I can take any e to a number, that domain was all reals. I really have to worry about the denominator not equal 0, so denominator is not allowed to equal 0, because if I get that, I'm dividing by zero and breaking every rule in math, nobody is happy with that. So I gotta find the values of x where 1- e to the 1- 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 it now becomes a good old solve for x and this should feel like a little pre calculus problem, we have to solve for x with e in the base. So let's move things over and we get e 1- x squared equals 1. So how do I solve for e? Let's take ln at both sides, it's good reminder of what the natural log does. Hopefully you seen this before. Pen is acting funny, sorry. So let me write this over. ln e to the 1- x squared is equal to ln of 1. Logarithms, reminder, friendly reminder, what is the natural log of 1? That's 0. Talk more about that in a minute and the reason why you take ln and e to both sides is because they cancel each other out, and the exponent falls down and you get good old 1- 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 1. But really the answer is plus or minus 1. So there's two values here that are not allowed, plus 1 and minus 1. 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 1. That's the set way to do it. That's fancy, you can also say from minus infinity to -1 with a parentheses, union -1 to 1, union 1 to infinity. Right, so I union, and I use open parentheses, that means don't include 1 or -1. 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. All right, 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 to the x, switch it up. Let's find the difference quotient. Remember this? Find the difference quotient, f(x + h)- f(x), all over h, this is an expression that will come up later. But for right now, it's just the complicated expression, manipulate the function, do something to it. And again, pause the video if you're working on this at home, try to plug it in and then check with the answer. Okay, here we go. This says take your function and where we see an x, plug in x + h. Since becomes 5 to the X plus H, take out the X plugin X plus H don't need parentheses here, but usually it's not too bad. Then minus the function, so minus 5 to the X. Over h what's H, don't know some variable? Why's it there? Because it's there, I don't know. Okay, is there anything I can do with this? Is anything cancel doesn't look like it. There isn't too much. In all honesty 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 to the X times 5 to the H. Right, so this constant, the rules of algebra. If you have the same base multiplied together, you add the exponents and kind of doing it in reverse here everything else stays the same. And the reason why I do this because we will use this form later. There's a 5 to the X in both pieces, so if I factor that out, I get 5 to the 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 will work with this later, but for now this is the difference quotient. With this one, let's do one more example. One more example and then will stop here and let you guys go do some on your own so true or false. I'm going to give you a function a different than we saw, although similar. This function 1 minus E to the 1 over X 1 plus E. To the one over X is an odd function. Now a lot of people get this wrong, mostly 'cause they forget what it means to be an odd function friendly reminder. An odd function is if you have origin symmetry, origin symmetry. Now 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 kind of 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. Doubt many of us can do that in our heads. We have to work this out, so you shouldn't look at this in to say, It's obviously true or obviously false. If you want to sound fancy and hedge your bets 50 50 shot. But let's see what happens if we work this out. Okay, here we go. So let's plug in everywhere that I see in X. I'm going to plug in. A minus X. And that's to all spots numerator and denominator. Okay, So what is that? What's going on with that? How do I clean this up? Each one negative exponent. You say is there anything you can do? Yeah, when you have a negative exponent, remember the rule. If I have like A to the minus N, it becomes 1 over 8 to the negative exponents become positive in the denominator. So let's do that. Let's write this as 1 over E to the 1 over X. And then 1 plus E to the 1 over E to the 1 over X. Ignore that. So negative exponents become fractions. Now I took it complicated expression and made it into a complicated fracture. I have 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 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 either the 1 over X, either 1 over X. Let's write 1 with a common denominator. 1 over E, the 1 over X. That's my big fancy numerator all over. E to the 1 over X, E to the 1 over X. So that's just a fancy way to write the number 1 plus 1 over E to the 1 over X. So at this point, and I 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 either 1 over X is in here that I'd like it to be true. Or at least I like to see something happened to tell me if it's true or false either way anyway, so let's add, subtract and add fractions. How does that work? Well, I have comma nominator. So I add the numerators and you get E to the 1 over X minus 1. Over E to the 1 over X. And then that becomes E to the 1 over X plus 1. Over. E to the one over X, so fraction divided by fraction. And say, all right, what do you do now? Well, this is a crazy exercise in algebra. You can keep change flip, right? You can multiply the top and the bottom by e to the one over x. And let's do that. So keep change flip. We keep the numerator into the one over X minus 1. E to the 1 over X, division becomes multiplication and the nominator flips e to the 1 over X. And you get e to the one over X +1. And now all of a sudden, because I'm multiplying, these to the 1 over X actually do cancel. Now 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 to the 1 over X. Now 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 we 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 test 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. Okay, this is a tough one just because of the algebra involved. Go over this example any 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.
