Welcome back, everyone. By now, we've worked with a ton of different polynomial functions, and one of the most common and perhaps most basic polynomial functions we've seen is f(x)=x2. But if I take that x and that 2 and I swap them, I now have f(x)=2x, which is an entirely different type of function called an exponential function that we're going to have to evaluate and graph just as we have other functions. Now I know seeing that variable in the exponent might seem a little bit strange at first, but here I'm going to walk you through the basics of exponential functions using a lot of what we already know about exponents themselves and what we know from working with other functions. So let's go ahead and get started.

Now looking at our function here, 2x, we have two different things going on. We have this base number of 2, and then we have the power that that base 2 is raised to, in this case, x. When working with exponential functions, we need to consider a couple of different things about the base of our function. In this case, we have 2. But the base of any exponential function needs to be constant, so it can't be something with a variable that changes. It also has to be positive, so we can't have any negative numbers as our base. And it cannot be equal to 1. Thus, something like 2 fits all of that criteria.

Now when considering the exponent or the power of our exponential function, we only need to consider one thing, and that is that it contains a variable. So it's something that can change, something like x. Now that we know a little bit about exponential functions, let's take a look at a couple of different exponential functions down here. So here we have this function f(x)=23x. And we want to determine if this even is an exponential function. If it is, then we can go ahead and identify both the power and the base.

Looking at this function here, I have 2/3 as that base. I want to consider my three things: constant, positive, and not equal to 1. So 2/3, it is constant, it is positive, and it is not 1. So it looks like we're good so far. Now looking at our power here, we have this power of x. We need to make sure our power contains a variable, which it already is a variable just by itself. So it looks like we are good here, and this is an exponential function.

Now to identify our power and our base, our power of x there, and then our base is 2/3. It's okay to have a base that's a fraction as long as it's constant and positive. Let's move on to our next example.

Here we have f(y)=1y. Now our base here is 1. 1 is constant, it is positive. But it is 1, so it does not meet that last criteria of not being equal to 1. So I don't even have to look any further. I already know this is not an exponential function. 1 to the power of anything is just going to be 1, so that power of y doesn't even matter, which is why this isn't an exponential function at all.

Let's look at one final example here. Here we have 10x+1. And we want to identify if this is an exponential function. Here my base is 10, so this 10 is constant, it is positive, and it is not 1. So we're looking good so far. Now looking at our power here of x+1, it does contain a variable; it has that x. So it looks like, yes, we are dealing with an exponential function here.

Now here our power is this entire thing x+1, it's not just the variable by itself; it's everything that our base is being raised to. Now here, our base is this 10 because that's what's being raised to the power of x+1. So we are good here with our power x+1 and our base of 10.

Now that we know how to identify an exponential function, let's go ahead and get into evaluating them. Now, we're going to have to evaluate exponential functions for different values of x, which just means we're going to plug in values of x to our function. So here we have f(x)=2x. So to evaluate this for x equals 4, I'm simply going to plug 4 in for x. So here, f(4) is really just 24. Now using my rules for exponents, I know that 24 is really just 2 times 2 times 2 times 2 4 times, which will give me 16 as my final answer.

Let's move on to our next example here and evaluate our function for x is equal to negative 3. Now, plugging 3 in for x here, f(−3) gives me 2−3 because we're still using that same function here. Now whenever I have a negative in the exponent, that's totally fine. It just means that I'm really dealing with 1 over 23 because it's really a fraction.

Now with this 23 on the bottom, this is really just 2 times 2 times 2 3 times. So my answer here is simply 18. I end up with this fraction. Now let's move on to our next example. We have f(x)=3.14. So, of course, plugging that 3.14 in for x to our function, we have f(3.14)=23.14. Now this is not something that I want to or I may have been capable of doing by hand, so this is when I would actually want to use my calculator.

So if you're ever unsure of how to do it by hand, just type it into your calculator. Now the first thing we're going to do is type in our base, in this case, 2. Now in order to raise something to a power to get that exponent, we're going to use the caret key that looks like this on your calculator. So I'm going to take 2, raise it to the power using that caret key, and then I'm going to simply type my power in, in this case, 3.14. Now you can always add some parentheses around everything if you just want to make sure that everything is going into your calculator the way that you mean it to. So if I type this in my calculator, I take 2 and raise it to the power of 3.14, I end up with about 8.815 as my final answer.

Now let's take a look at one final example here. Here we have x is equal to 12. So in order to plug 12 into my function for x, I can go ahead and take f(12) and that gives me 212. Now I don't really want to multiply 2 by itself 12 times. So if your exponent is rather large and you don't want to do it by hand, go ahead and type that into your calculator. So here we would do 2 caret 12. And in our calculator, we would end up getting 4,096 as our final answer.

Now that we have a good idea of what to do when working with exponential functions and how to evaluate them, thanks so much for watching and I'll see you in the next one.