Exponential Functions - Online Tutor, Practice Problems & Exam Prep

Exponential Functions

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. 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 2. 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 1. So 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 like x.

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 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 23 as that base. And I want to consider my three things constant, positive, and not equal to 1. So 2 thirds, 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, we saw our power of x there, so our power is simply x by itself.

And then our base is 23. Now 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. So here we have f(y)=1y. Now our base here is 1. 1 is constant. It is positive, but it is 1, so it is not meeting that last criteria not being equal to 1. 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 in my base. 1y 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. So here my base is 10. 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+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 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 f(x)=2x.

So to evaluate this 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 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 wi

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

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

Determine if the function is an exponential function.  

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

$f\left(x\right)=\left(\frac12\right)^{x}$

• A. Exponential function, $f\left(4\right)=\frac{1}{16}$
• B. Exponential function, $f\left(4\right)=-16$
• C. Not an exponential function

Graphs of Exponential Functions

Hey, everyone. Whenever we graphed polynomial and rational functions, we had to consider a lot of different components. We looked at things like the end behavior and any intercepts of our graph and even had to consider and find multiple asymptotes sometimes when working with these functions. Now that we're faced with graphing exponential functions, you may be worried that there's going to be a lot of new and different things that we have to remember in order to fully graph them. But here we're going to walk through graphing the exponential function \(2^x\) together, and I think you'll find that it's actually even more simple and more straightforward than graphing any polynomial or rational function.

So let's go ahead and get started here. Now here we're going to be graphing \(2^x\). So let's start by just plugging in some values for \(x\) and getting some ordered pairs to plot on our graph. So starting with \(x = 0\), if I plug that into my function, I get \(2^0\), which is simply 1 because anything to the power of 0 is 1. So that first point I can plot is right at 1.

Then moving on to \(x = 1\), plugging that into my function, I get \(2^1\), which is simply 2. So that next point is at 12 that I can plot on my graph. Then as \(x\) continues to get larger, I'm gonna get \(2^2\), which is simply 4, and then \(2^3\), which is 8. And if I took it all the way up to \(x = 10\), I would get an even larger number, this 1,024. So I can see that as \(x\) continues to increase, \(f(x)\) will continue to increase as well.

So let's plot those last two points at 24 and at 38. Now looking at this side, I can go ahead and connect all of these points because I know it's going to increase. So I see that as \(x\) goes to infinity, \(f(x)\) also goes to infinity. That's my end behavior on that side. Now let's consider if \(x\) is negative.

So starting with \(x = -1\), if I plug that into my function, I get \(2^{-1}\), which knowing our rules for exponents, this is simply going to be 1 over 2. So that point is at negative one, 1/2. And then as \(x\) gets more negative, -2 and -3, it continues to get smaller and smaller on that side. So we see that we're gonna get really close to 0, but not quite 0. So I can go ahead and connect my points on this side.

I see that I'm getting really close to that \(x\) axis there. So as \(x\) approaches negative infinity, I see that \(f(x)\) actually approaches 0. Now something I said there might have sounded familiar. We're gonna get really close to 0 but not quite touch it because that's what we know as an asymptote. So we do have an asymptote.

It is a horizontal asymptote right on that \(x\) axis at \(y = 0\). Remember, we can plot asymptotes by using a dashed line. So here I have my dash line right at \(y = 0\). Now taking a full look at our graph here, looking first at this right side, I see that as we increase, we look kind of similar to what we've seen with polynomial functions. Now another similarity that our graphs of exponential functions have to polynomial functions is that they are always going to be continuous, meaning that there are no breaks in our graph.

So we see that there are no breaks in this graph. It is continuous. Now looking at the other side of our graph here as we approach that asymptote, we know that we have asymptotes when dealing with rational functions. So another similarity that we may have to rational functions here is that our function is going to be 1 to 1 the way that some rational functions are. Now this just means that no 2 \(x\)'s result in the same \(y\) or we could pass the horizontal line test.

If I draw a horizontal line, it's only going to cross through one point on my graph. Now looking at the complete shape of the graph of my exponential function, this is always going to be the shape. And a way that you can remember this is exponential function starts with \(e\). So if I take a lowercase \(e\) and I just extend that tail out, it looks really similar to the shape of this graph. Now that we have a complete picture of this graph, let's also consider the domain and range here.

So the domain of any exponential function is actually always going to be the same. It's always going to go from negative infinity to positive infinity, or simply all real numbers. This is the same for any exponential function. Now your range is a little bit more complicated but it just depends on your asymptote. So it's always going to depend on the placement of our asymptote here.

And here my asymptote is at 0. And I see that all of my \(y\) values are above that. So I know that my range here is going to go from 0 to infinity, but not including 0 because I am, of course, not touching that asymptote. Now that we have a good picture of our graph of \(2^x\), let's take a look at some more exponential functions. Now as we look at different exponential functions of the form \(f(x) = b^x\), we're going to see that the direction and the steepness of our graph is going to depend on the value of \(b\), the base of our function.

So looking at if \(b\) is greater than 1, like something like 2 or 5, I'm gonna see that whenever I have that \(b\) greater than 1, my graph is always going to be increasing and going up on that side. Whereas if \(b\) is between 0 and 1, so it's something like 1/2 or 1/4, I'm instead going to have a graph that decreases. Now, of course, also the steepness will depend on this value of \(b\). And here we consider \(b\) greater than 1. You see that as we increase from 2 to 5, our graph gets steeper.

And that's because for values of \(b\) are base that are greater than 1 like 2 or 5, our graph is going to get steeper for larger values of \(b\). Now it's actually the opposite when working with values of \(b\) between 0 and 1, those fractions, where we see that as we go from 1/2 to 1/4 on our graph, as we get smaller, it actually gets steeper there. So for values of \(b\) between 0 and 1, our graph is actually going to get steeper for smaller values of \(b\), so the opposite of if \(b\) is greater than 1. So now that we have a complete picture of the graphs of our exponential functions, let's get some more practice. Thanks for watching and let me know if you have any questions.

Graphing Exponential Functions Example 1

Hey, everyone. In this problem, we're going to work through graphing the function f(x) = 1/2^x together. So let's go ahead and get started. Now we're just going to go through and plot some points by plugging in some values into our function. Let's start with x = 0 and plug 0 into our function and see what we get.

So if I take 1/2 and raise it to the power of 0, I know that anything raised to the power of 0 is 1, regardless of if it's a fraction or not. So my very first point is at (0, 1), and I can go ahead and plot that on my graph. Then if I plug 1 in for x, I get 1/2¹, which if it's raised to the power of 1, it just gets itself. So this is simply 1/2, and I could plot my second point at (1, 1/2). Then for x = 2, plugging that into my function, I get 1/2².

Now 1 raised to the power of 2 is just 1, so really, I'm just doing 1 over 2², which is going to give me 1/4. So we're getting much smaller here, (2, 1/4), getting close to that x-axis. And then finally, for x = 3, if I put 1/2 and raise that to the power of 3, that's going to give me 1/8, which is again getting really small and close to that x-axis over here. Now looking at our negative numbers, if I start to plug in negative numbers for x starting with negative one, if I take 1/2 and raise it to the power of negative one, whenever we deal with whole numbers like 2 and raise them to a negative power, we end up with a fraction. Whenever we take a fraction and raise it to a negative power we are instead going to end up with a whole number so this is simply going to give me 2 because I just flipped that fraction I get 2 over 1 that's just 2.

So I can plot this point at (-1, 2). Then if I take 1/2 and raise it to the power of negative 2, I'm going to get 4. So I can plot this point at (-2, 4). Then for my last negative point, if I take 1/2 and raise it to the power of negative 3, I end up with 8, and I can, of course, go ahead and plot that point on my graph. Now we see that we have all of these whole numbers as we get into the negatives, and then we had their corresponding fractions.

We just flipped them under to the bottom of a fraction, and they're the same numbers because we're dealing with the same powers. Some are positive and some are negative. So some of these values will be fractions and some will be whole numbers, just like we saw for 2^x. Now, if you think about it, 1/2^x is really just 2^-x. So that's why we're seeing so many similarities here.

Let's go ahead and connect our graph. So on this side, going up. And then on my right side here, I see that I'm getting really close to that x-axis, not quite touching it, though, which tells me I'm dealing with an asymptote. So I can go ahead and plot my asymptote here on this x-axis because it is a horizontal asymptote right at y = 0. And we can go ahead and denote that here, our asymptote at y = 0, plotted, of course, using a dashed line because it's an asymptote.

Now looking at our graph here, you might notice that it looks really similar to the graph of 2^x which looks something like this. So now our graph of 1/2^x is kind of just this flipped which makes perfect sense because we just said that this 1/2^x is the same exact thing as 2^-x. So it makes sense that it's the same graph but flipped because it just underwent a transformation due to this negative sign here. Now let's go ahead and finish up here and get the rest of the information for this graph like the domain and the range. So here our domain of every single exponential function is always going to be the same.

So here it is the same as before: all real numbers. Now our range is dependent on where our asymptote is and since our asymptote is again at y = 0 and our graph is completely above that, we know that our range is going to go from our asymptote at 0 to infinity. So our range is (0, Infinity) using parentheses because that 0 is not included. Now that we have the entire picture of our graph of 1/2^x, let's get some more practice.

The Number e

Hey, everyone. When working through problems with different exponential expressions and exponential functions, you may come across one that doesn't have a base of something like 2, 1/2, ten or any other number, but actually has a base of e, this lowercase e. And the first time you see that, you might be wondering why there is another letter in that function when we already have x right there, and how we are going to work with this function with 2 different variables. But you don't have to worry about any of that because here I'm going to show you how e is literally just a number, and we can treat it just like we would any other exponential function, and evaluate and graph it using all of the tools that we already know. So let's go ahead and get started.

Now, like I said, e is not a variable at all but simply a number, and another similar number that you might be a bit more familiar with is pi. We know that pi is 3.1415 and so on, this long decimal that we don't write out. We just write pi. E is really similar. It's this long decimal, 2.71828 and so on, but we simply write it as e.

Now because it's a number, we can treat it just like we would any other exponential function and do things like evaluate it for different values of x. So let's take a look at our function here, f(x)=ex, and go ahead and evaluate this for x equals 2. Now I'm simply going to plug 2 in for x into my function, so f(2) is going to give me e2. Now when working with exponential functions with base e, we do want to use a calculator to evaluate these. And the buttons that you're going to use on your calculator in order to get this base e are second ln.

This should give you e raised to the power of and then you simply type in what your power is. So for e to the power of 2, I would type second ln and then 2 in order to get my answer, rounding to the nearest 100ths place, which would be 7.39. Now this would be my final answer here, but let's go ahead and evaluate this function for x equals negative 3. Now, again, we're just going to be plugging in negative 3 for x here. So f of negative 3 is simply e-3, which knowing our rules for exponents, this would really just be 1e3.

And you can type either of these into your calculator, and you should get the same answer. So typing e to the power of negative 3, I would type second ln and then negative 3, and rounding to the nearest 100ths place, I would get an answer of 0.05 as my final answer here. Now, we can evaluate exponential functions of base e like any other exponential function, and we can also graph them just like we would any other exponential function as well. So if I take my function here, e to the power of x, I know that my graph is going to have the exact same shape as any other exponential function.

So my graph is going to end up looking something like this, f(x) is equal to ex. And you see here that it's right in between my graphs of 2 to the power of x and 3 to the power of x, which this happens because e, we know, is this number, 2.718 and so on, which happens to be right in between 23. So I know that my number e is right in between 23, so it makes sense that the graph of my function e to the power of x is right in between the graphs of the exponential functions with base 2 and base 3. Now if you're faced with graphing a more complicated function of base e, you can simply graph it using transformations, the same method that we use for any other more complicated functions of different bases like 2 or 3. Now hopefully, with all of this, you see that we can treat exponential functions of base e just like any other exponential function no matter what the scenario is.

But you still might be wondering why we need this base of e in the first place. Why do we need to have this base e when we have all of these other numbers to choose from that aren't crazy decimals? So I'm going to give you a little bit more information about what e is and where it comes from. So e actually comes from the idea of compounding interest, which is this equation right here. And we want our interest to compound as much as possible.

So if I take the number of times that my interest is compounded, this n, and take this all the way up to infinity, this equation is going to end up giving me 2.71828 and so on, which we know is just e. Now that's where e comes from, compounding interest, but it's actually going to pop up in a ton of other stuff that you'll see. Now you might see in your other courses that e is a part of predicting population growth and working with something like radioactive decay and half-lives. So e is just a number, but it's a number that describes a bunch of different things going on in the world. And even with describing all of these different things and being super useful, we can treat it just like we would any other exponential function.

So with that in mind, hopefully you have a better idea of what e is and why we need it, and how exactly to work with it. Thanks for watching, and let me know if you have any questions.

Graph the given function.

$g\left(x\right)=e^{x+3}-1$

• A.
• B.
• C.
• D.