Introduction to Exponential Functions - Video Tutorials & Practice Problems

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Exponential Functions

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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 of X is equal to X squared. But if I take that X and that two and I swap them, I now have F of X is equal to two to the power of X, 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 two to the power of X, we have two different things going on. We have this base number of two and then we have the power that, that base two is raised to. In this case X. Now, in working with exponential functions, we need to consider a couple of different things about the base of our function. So in this case, we have two, but the base of any exponential function needs to be a constant. So it can't be something with a variable that changes, it has to be positive. So we can't have any negative numbers as our base and it also cannot be equal to one. So something like two 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 of X is equal to two thirds to the power of X. And here we want to determine if this even is an exponential function. And if it is, then we can go ahead and identify both the power and the base. So looking at this function here, I have two thirds as that base. And I want to consider my three things constant, positive and not equal to one. So two thirds, it is constant, it is positive and it is not one. So it looks like we're good. So far. Now, looking at our power here, we have 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, we are good here. And this is an exponential function. Now to identify our power in our base, we saw our power of X there. So our power is simply X by itself and then our base is two thirds. Now it's OK to have a base, that's a fraction as long as it's constant and positive. Let's move on to our next example. So here we have fa Y is equal to one to the, to the power of Y. Now our base here is one, one is constant, it is positive but it is one. So it is not meeting that last criteria, not being equal to one. So I don't even have to look any further. I already know that this is not an exponential function. I don't need to worry about identifying my power and my base one to the power of anything is just going to be one. So that power of why 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 10 to the power of X plus one. And we want to, if this is an exponential function. So here my base is 10. So this 10 is constant, it is positive and it is not one. So we're looking good so far. Now, looking at our power here of X plus one, 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 plus one. 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 plus one. So we are good here with our power of X plus one 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 of X is equal to two to the power of X. So to evaluate this for X equals four, I'm simply going to plug four in for X. So here F of four is really just two to the power of four. Now using my rules for exponents, I know that two to the power of four is really just two times, two times, two times 24 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 three. Now plugging three in for X here, F of negative three gives me two to the power of negative three because we're still using that same function here. Now, whenever I have a negative in the X, that's totally fine. It just means that I'm really dealing with 1/2 to the power of three because it's really a fraction. Now, with this two to the power of three on the bottom, this is really just two times, two times, 23 times. So my answer here is simply 1/8. I end up with this fraction. Now let's move on to our next example, we have FX is equal to 3.14. So of course, plugging that 3.14 in for X to our function, we have F of 3.14 is equal to two to the power of 3.14. Now, this is not something that I want to or I'm 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 too. Now, in order to raise something to a power to get that exponent, we're going to use the carat key that looks like this on your calculator. So I'm going to take to raise it to the power using that carat key. And then I'm simply going to 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 two 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 head and take F of 12 and that gives me two to the power of 12. Now, I don't really want to multiply two 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 do two carat 12 and in our calculator, we would end up getting 4096 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.

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Problem

Problem

Determine if the function is an exponential function.

If so, identify the power & base, then evaluate for $x=4$.

$f\left(x\right)=\left(-2\right)^{x}$

A

Exponential function, $f\left(4\right)=16$

B

Exponential function, $f\left(4\right)=-16$

C

Not an exponential function

3

Problem

Problem

Determine if the function is an exponential function.

If so, identify the power & base, then evaluate for $x=4$ .

$f\left(x\right)=3\left(1.5\right)^{x}$

A

Exponential function, $f\left(4\right)=410.06$

B

Exponential function, $f\left(4\right)=15.19$

C

Not an exponential function

4

Problem

Problem

Determine if the function is an exponential function.

If so, identify the power & base, then evaluate for $x=4$ .