Hi, everyone. Welcome to our lecture on exponential functions. Just to remind you what we've seen so far, we've seen functions of the form f of x equals x squared. These are called polynomials, these are quadratic functions, their graphs are parabolas. Notice the variable is in the base and then the constant, the number is upstairs. What we'd like to do now is switch it and talk about exponential functions, and we're going to write things now the form 2^x, or perhaps 3^x, 4^x. You can pick any number you want as the base, but the key is the base is now a constant and the variable now becomes the exponent. When the variable is in the exponent, these are called exponential functions. The base can be any number you want, it doesn't have to be a pretty number like two or three, it can be Pi or square root of 2 or something like that. This leads to our general definition of an exponential function. We say an exponential function with base a, we're going to let a be some constant here is of the form f of x equals a^x. Again, notice that even though there's letters everywhere, a is a constant and the variable is in the exponent. X can be any real number with no restriction on x, but we have some restrictions on the base. We need the base to be positive and a is not equal to 1. We'll talk more about these restrictions in a second. Let's look at some examples here. Again, you can graph these on a calculator or you can plug them into Desmos if you'd like as well. Let's take the function f_x equals 2_x. Start plugging in some values here, notice our base here is a. If you plug in x equals 0, you get 2_0, which of course is 1, and if you get 2_1, that's 2, and then 2 squared is 4. It starts to grow very quickly and this behavior, this graph looks like exponential growth. This is the general shape for any value of a that is greater than 1. We're going to look at exponential growth. In the case that a is 1, let's think about this for a second. Why would I not want 1? There's nothing stopping me from doing 1_x, but what's the problem? What's 1 squared? What's 1 cubed? What's 1_4? Instead of an exponential function with this nice exponential shape of a graph, I just get a constant function, I just get 1 for everything? This is not really an exponential function, this is actually the function f of x equals 1. Because it's not really an exponential function and I don't want to include it, we throw it out. We say, "Look, if you want to talk about constant functions, go do that somewhere else but I don't want to do it here when I'm trying to study exponential functions." The other case that you can have is if a is a nice little fraction between 0 and 1. Let's say 1/2 to a power, something like this, that's perfectly allowed. Again, I'm positive and I'm just less than 1. In that case, think of the numbers you're going to get. Graph this if you want to think about it. If you plug in 0, you still get 1, that's all fine, but then all of a sudden if you take 1, you get a half, and if you plug in 2, you get a fourth, and your numbers get smaller, and that is going to be the general shape. We're going to say this is exponential decay when you plug in as a base, a number that's less than 1. Because in that case when you start taking powers, the numbers get smaller. Now, you can see this is 1_x by properties of algebra. Remember, the exponent hits both the numerator and the denominator, so honestly it's more common to see this as 1 over 2x. Again, if you see it in this form, just remember the base itself is 1/2 and not just 2, they brought the x into both the numerator and the denominator. Whenever we have a new function, we want to start asking general questions that we would ask for any function including what's the domain. What are the numbers that I'm allowed to plug-in to a function of the form f of x equals a_x? Just remember that a is positive and a is not equal to 1. What am I allowed to plug in here? It turns out that the domain of this function whether you have growth or decay, so as to remind yourself what the graphs look like in general. This is the graph where a is greater than 1, and if you have decay, we go from high to low. This is the case where I'm between a equals 0 and 1. I have growth or decay. In any case, in either of these two cases, the domain turns out to be all real numbers. There's not a problem plugging in any power you want. This is any real number you want. Let's say I just take the easy case 2_x, something simple. But what does it mean to plug in different numbers? If I take f of 4, this becomes 2_4. That's fine, that's 16. I think there's nothing wrong with doing that. What if I take something that's not so pretty? What if I take Pi? Well, you have 2_Pi. Now, we don't know that number, but your calculator does. This is just some number, when we write it, we usually leave it as through the Pi, if the question never wants you to write a three decimals, four decimals, or whatever, then sure go grab a calculator and plug it in. But pick your favorite ugly numbers and plug-in and it will all work. If you need decimals again, grab technology, grab a calculator, grab the internet, but you can always write these things. The domain is all reals. The next question, of course, is what is the range? If you look at the graph of either case, either growth or decay, the outputs of all these functions are positive and you have a nice horizontal asymptote in either case. The range in this case we're going to say is all y values that are positive. I write y is greater than 0. Notice I do not include 0, this function never gets to 0. We have a nice asymptote at the x axis, so y is greater than 0. There's a few ways to write this range. You can write it using inequalities, of course, you can also write this as 0 to infinity. Notice I'm using parentheses on both 0 and infinity to show that I don't want 0, and of course you always use parentheses with affinity because you can never get to infinity. There's lots of ways to write this. If you want to get super fancy, of course, you could also write this as real numbers, but just the positive ones. Sometimes, once in a while you see this is R plus. For any exponential function in it's base form, f of x equals a_x, you're going to have domain all reals and range all positive numbers. This will change if you start translating or rotating or shifting the graph, but in it's basic form, this is your domain, this is your range. No matter what kind of exponential function you start with, whether it is growth or decay. Friendly reminder, if you have growth, we say it goes from low to high, this is when your base is greater than 1. If you have exponential decay, think like zombie apocalypse, then you say that a is less than 1, but by restriction it has to be greater than 0. You're in this very specific case. Now, notice they're still infinitely many numbers between 0 and 1. There's lots of numbers you can choose from. We have growth or the decay. There's some common themes to both of these and these are called properties of the exponential function. The first one that says, "If a is greater than 1 then the function is always increasing." This is a very nice property, it goes up forever. It never has a hump, it never goes down and never starts to hit decrease, it's always increasing. If a is in the other case where I'm between 0 and 1, then f of x is always decreasing. This is interesting. It's a function, I don't know if we've seen these before, but it always goes up or it always goes down and there's no chain. In particular, they all have very nice intercepts. The y-intercept, remember how to find the y-intercept? We set x equal to 0 and no matter what your base is, if I have a to the 0, for the reminder any number raised to the 0 is going to be 1. The y-intercept of the graph is always at 0, 1. Oftentimes, if you start picking random points on the curve, they're pretty ugly numbers. They're usually decimals or gross. But every single graph in this form a^x is going to have a very, very nice intercept. When you have nice numbers that appear on the graph, you should know what they are. The y-intercept is a nice one again, that's because if I plug in 0, sometimes people confuse this like, "What's 2^0?" Everyone just say 1. "What's 3^0?" 1, those sort of things. Next, number 3, is that the graph has a horizontal asymptote. I'm gonna abbreviate that with HA, horizontal asymptote at the x-axis, which is given by the equation y equals 0. Either growth or decay, There's a nice asymptote here. Remember what our asymptote is. It means that the graph is going to get infinitely close, but never touch. We have a nice horizontal asymptote in either case. Four, no matter if we're growth or decay, the domain is going to be all reals and the range is going to be positive y-values. Last but not least, the function f of x is one-to-one. Now, let's talk about one-to-one for a second. One-to-one means we pass the horizontal line test. Horizontal line test says if you start drawing horizontal lines, then you're only going to cross the graph once. That means that for every input, there is only one output. The algebraic way to say that is, if f of x_1 equals f of x_2, then that implies that x_1 equals x_2. This says if the y values are equal, if you have two different points and the y values are equal, the x value is the same. Again, compare this to the parabola where that's not true. If I take the y value at positive 1, I could have plugged in either positive 1 for x squared, 1 squared is 1, or it could have plugged in negative 1 because negative 1 squared is also 1. So there's two x values that go to the same y-value. This is just another property that separates this exponential graph and gives it its own characteristics, its own unique flavor. In particular, we will see that when we have a one-to-one function, we're going to have an inverse function. We're going to be able to define an inverse function. Maybe you know this, maybe you don't, but the inverse of the exponential, of course, is the logarithm. Step 5 here, property 5 is allowing us to pave the way so we can talk about logarithms, whether you want to or not. Solving exponential equations sometimes can be tricky, but we're going to use this one-to-one property of it. Here's the property that I want to use. If a^x_1 is equal to a^x_2, then x_1 equals x_2. Let's think about that for a second. Remember, the key here is the base is the same. If I have 2^3 equals 2 to the something, if I'd know the bases are the same, from that, I can write the equation. Three must equal my exponent. If the bases are the same, the exponents are the same. This feels like what we've been doing the algebra all along. If we do the left side, we must do to the right side. It's the same idea. For numbers, they're not too exciting. Obviously, 2 cube is 2 to the something, but it's often going to come into play when we talk about exponents. For example, if I have 2^3_x minus 1 is equal to 1, well, what do I do in this case? How do I solve for x? This is where we're going. How do we solve for x? Well, the key to this one is to realize, one, well, wait a minute, that's my y-intercept. So I can write this as 2, 3_x minus 1 is going to equal 2^0. Do we all agree that 2^0 is 1? I hope so. Now we're going to use the properties of exponential equations. I have the same base and since they're equal, I must have the same exponent. This is a beautiful little skill here because you take a very difficult exponential equation and you completely remove it. From this, you get, well, if the bases are the same, guess what? The exponents have to be the same and you have a nice linear equation. You've taken all of the difficulty out of this equation. From here, it's an easy sell for x. You get one-third and you're done. You can go back and check that it works, of course, in the original equation. This property of exponential equations is going to allow us to solve for x in a whole new family of functions. Let's do a couple of examples. Let's look at the example f of x equals 3 raised to the 4 minus x. I'd like to find the value of x such that f of x is equal to one-ninth. Let's say it one more time. When f of x is 3^4 minus x, find x such that f of x is equal to 1 over 9. This is in fact saying, let's set one-ninth equal to 3^4, minus x. Now, in this current form, this looks a little scary, but I want to use the property of exponential equations. In particular I really need things to be the same base. I see on-ninth. Now 9 is the nice power of 3. So why don't I write one-ninth as 1 over 3 squared? We're getting there. But that's still not a form of the power of 3, at least not yet. Do you see how I can turn this into a power of 3? Rule are we going to use to get this to be a power of 3? Well, 1 over 3 squared is the same as 3 to the minus 2. All of a sudden, I have two equations with the same base. This takes a little bit of rearranging. Don't be afraid to get in there and get your hands dirty as you do this. But 3 to the negative 2 is 3^4 minus x. That's amazing because now I took this exponential equation and I'm just going to look at the exponents. Because the bases are the same, I can leave the world of exponential equations and look only at the exponents, make them equal to each other, and of course, solve for x. Let's solve for x here. Let's add x to both sides. We get x minus 2 is 4, add 2 to both sides and I get x is 6. Even when they don't look like they have the same base to start with, look for powers. Oftentimes, when these questions are set up correctly, when you have 3, you'll have 9 or 27, or just some other powers that you'll find and you can write this. We'll get into more complicated exponential equations later. But for now, be able to solve these and I think you're off to a good start. Great job on this video and we'll see you next time.
