So how about learning the exponential function? But I can already see that you are asking, Gis, what am I going to use this for in my life? Do you ask your teachers this ?
Oh people, when students ask, oh my God, right, but anyway, why do you use an exponential function in everyday life? So see that it is used in various situations, what is it used for? To describe and model the behavior of these various situations and what are these various situations that I'm talking about?
Can you think of any? Let's see, in financial mathematics when I work with compound interest, in the growth of bacterial cultures there is a lot of exercise that we see in the growth of bacterial cultures and also in the behavior of cases of some diseases, right. Do you have anything else in mind?
Leave it in the comments for us to observe. So it is used in these contexts to describe the behavior of these situations. But now we have a definition for an exponential function that you need to know how to understand this definition too, so every exponential function will be like this one here, FX = A to the power of A to the power of X, these are notations , okay?
And now why is A raised to X? What does this have to do with here? So see that because it is called an exponential function, now the variable is in the exponent, that's why it is exponential, so you looked at some function and saw that in the exponent there is a little x there is a variable, it doesn't just have to be Just notation, but the variable is in the exponent so you will remember the exponential function, okay?
Then what happens then defined for all x Real so the the number greater than zero is the A that you see here which is our base, I just talked a little about X which is the exponent, the A which is the base it always has to be greater than zero it cannot be a negative number right, not even zero and it also has to be different from one so it can never be 1, okay? So those are the rules. You can find the definition written like this or maybe in your book the definition is like this FX = A to the power of How do I say this, this symbol, so I don't say something wrong, now I'm not sure of his name, he's saying that it can't be zero and that it has to be positive, okay, it's written here, I think this is easier definition and A different from 1, I'll look for it when the class ends, Ok.
So there's a definition there, great? Now some examples of how we have exponential functions. See that here in all these cases the variable is in the exponent in this case here we have the base equal to 2 here the base is a third here the base is 5 and here the base is 4/5.
Now see that you can also find an exponential function of this type here, so you will be left in doubt, for example 3 + 2 raised to the x, you can find a case like this, here it is of the type an exponential function, right, because we observe most In the exercises that talk about the growth of bacteria in bacterial culture, there is always a number added here to the power or there is a multiplication, so they are of the exponential function type, okay? Production is saying here that for exponential function there are also examples from computer science, is that production? This is what we're talking about here, can you think of any other different examples that we didn't talk about here in class, but now pay close attention to one thing, so here it's an exponential function type, right ?
This one here, the student might get confused, maybe X to the power of 2 because this isn't an exponential function? because oh, but there is x there Gis, but the x is in the exponent? Look carefully, X is not in the exponent, it is in the base, so this here is not an exponential function, this here is a quadratic function, okay, pay attention to this and For those who are asking, can the variable here be a negative number?
What we said here is for all x Real so I could put here 2 to the minus x, whatever, so I can say yes for those who are asking. And now we're going to look at the numerical value of an exponential function. So here numerical value of an exponential function: given the exponential function f x = 3 to the power of x calculate F of four.
What does it mean to calculate F of 4? Do you see that here, in our formation law, it wasn't x? Here it was also x, right, what happened here, I didn't change the X for four, that's not it, so I'm going to change this x here for four.
So see what it's going to look like, it's going to be 3 to the power of 4, look here, 3 to the power of x I replaced it with four, it's three to the power of four, that's all? That's all folks, so now what is 3 to the power of 4? It's 3 X 3 X 3 X 3 and it's not 3 So here it will be 3 x 3= 9, 9 x 3= 27, 27 x 3= 81, is everything ok?
Quiet, right? Look at this F now when X is negative two then see that I'm changing I always tell students, it's very important to understand the content is to remember previous content, I'm working here with the concept of potentiation with this one also the potentiation plus with a negative exponent what you do remember well you put one below like it's the whole number and it inverts so it's 1/3, oh Gis but why does it invert? There's a class explaining why this inversion, okay?
I will leave a recommendation for you. The moment I invert the base the exponent becomes positive, so it will be 1/3 raised to positive 2. So it's going to be 1/3 x 1/3 this is going to be one times one because I multiply the numerators with each other and the denominators with each other.
So the answer to this here will be 1/9, okay? And if you have an exercise that says the letter C, I 'll add it here now because I want you to try doing another one, OK? Now I'll go like this, I just want to see fx = 27 and now what you have to do in this case is a numerical value, yes it is a numerical value although I have to find out in this case you can see that it gave that your function = 27 so you it will come here in its place, you will put 27, we are still talking about this exponential function, okay, I'm not leaving it.
And now how am I going to do it, how does it stay in place? So from my exponential function I put 27 because the exercise gave me is 27, it's = 3 to the x copy which is 3 to the x. So in this type of exercise here the numerical value is the opposite, right?
It gave you the value of the function and that is the value now of X, remembering that when I am working with the exponential function or any other function we always have the dependent variable and the independent variable x in this case the variable x and it is the independent variable Ok and the variable Y as I said at the beginning that I can change here it is the dependent variable, okay? Why are you dependent? Because the function, in order for me to find the value of the function, I depend on the value of X, Ok?
How do I finish this now? Here we have an exponential equation and to solve an exponential equation, we have to leave it with the same bases, so look, I'm going to do 27 here, oh Gis, how do I leave 27 in a base 3, just factor 27, so if you factoring 27 , 27 gives three, which gives nine times three, three times three gives one, so 27 is the same thing as 3 to the power of 3, so here, instead of 27, I'm going to write 3 to the power of 3 = 3 to the power of x good now that we have the same bases I can compare the exponents so I know that in this case x will be worth 3, take the real proof and come back here, if x is worth 3, 3 to the power of 3, how much will it give? value of function 27, ready, here, let's do it mentally, just by checking it out, right, but to explain the procedures here to you.
Easy when you need to calculate the numerical value of a function, but now for the student who is asking what if this were a fractional number, how would I do it? Let's go then. Now with the fraction I have the function fx = 1/15 raised to x.
So our base is now fractional, okay, calculate F of 2. So that means I have I want to change the 1/5 which will be 1 X 1= 1, 5 x 5= 25 right? Now see that x is worth minus three, so it will be 1/5 to the power of minus 3 and now how do I do this from here when a base is the exponent, which I always say base, the exponent is negative?
So you will invert this base which will be 5/1 but I don't need to put this 5/1 because 5/1 is 5 so just leave 5 as a whole, okay? It will be 5 and the exponent was positive 5 to the power of 3 it will be 5 x 5 x 5 this here will result in 125 and in that last case it is not similar to the previous one what is the value of x when our function is = 625? So see here, instead of this function I put 625, how will 625 be the same, instead of this one for those who don't understand very well, instead of this I put 625 equal to 1/5 raised to the x ok Gis, but now how I'm going to leave the base 625 and the base in fifth the same, which base can you work with here?
In base 5 and it is also very important for you who are watching this class that is in doubt about exponential equations, I have the classes, two separate classes on exponential equations in various cases , so take a look there. How do I do it here? Now I know that 625 is base five if you do the decomposition into prime factors of 625.
Look what will happen 625 gives by five which gives 125 by five 25 by five gives five and by five gives 1. So it's one two three four factors 5 to the power of 4 and this one fifth here we can invert this base too if you want or you invert this one you have to choose it will be the same way if I invert this one here goes staying one over 625 but the exponent is negative, is that ok? If you want to do it like this, then let's write here, 625, I know that it is 5 to the fourth power, 5 to the fourth power , which is equal to a fifth power to the become a fifth because I inverted this one here I came here and I inverted one, by doing this this exponent that was positive four becomes negative four equals 1/5 to the power of x o the bases are already the same I can compare the exponents so x = -4.
Ready, can we do the resolution and find the value of x in this case, okay? Did you understand well? And now we're going to talk about the types of exponential functions, shall we?
So see that there are two types of possibilities for an exponential function to occur, either it is increasing or it is decreasing. Okay Gis, but so what, what happens? See that every time you have an exponential function in which its base, which is a, resembles a, right, greater than one, a number being greater than one, you will have an increasing function, for example the one we talked about fx = 2 to the power of x you see here is greater than one the base fx = 1.
4 raised to x you see the base is greater than one so every time this happens you will have an increasing function so the decreasing one is when a is less than 1 it is not the People remember that a cannot be a negative number, it is greater than zero, it is very different from one. So every time you have a base between 0, between 0 and 1, what I just wrote here would be here for you to observe between 0 and 1, if you have a number here, a base that is in this range from 0 to 1, your function will be decreasing, for example, we just made one, so f of x = 1/5 to the x was not the size I made these parentheses, we just made one of these. Ah, but why is a fifth raised to the x, it's between 0 and 1, take it, divide the one by five, let's see who's okay, division 1 / 5 will be zero comma, right, two because 2 x 5= 10 so it's a number here which is between 0 and 1 so it is a case in which it will be decreasing f and will make of of x big again, it is a medium raised to x it is also a decreasing function and you can observe in some cases here be careful when you will classify because sometimes there is a prank he writes like this and there won't be a prank, right because zero put 0.
3 there, you will know that it will be a number between 0 and 1, it won't even be a prank, I'm trying to travel on mayonnaise here people, Alright, so about the types of the function, is it increasing or decreasing? Now I'm going to talk a little a little bit for you about the properties of the exponential function and then graphs, shall we go then? Look at this first property that we have here is f 0 = 1, what does this mean here in the exponential function, so for example if I put the function f of x = 2 raised to the x to the x you change the X to zero, what do you mean?
what will happen will be f of 0 = 2 raised to 0, and 2 raised to zero from the properties of powers that we have already studied, every number raised to zero results in one so we consider the first property of exponential functions. The second case that I didn't even mention here, look, hey, I wanted to see who is finding the error, the exponential function is injective or injective, if you want to talk about what it means, this means that for different values of x the image will also have different values in summary here o the image of x 1, that is, the image of X1 is different from the image of X2, remembering that X1 also has to be different from X2, it has to be, it won't be. Third case we just discussed right now about the graph that will be the function will have an increasing or decreasing graph and this graph will never cross the X axis.
It will always pass close to the X axis, we say that it will lean towards zero but It never cuts the X axis. And another observation that is not related to properties, but I didn't mention at some point that the inverse function of the exponential function, you know what it is, is the logarithm function. And finally let's go to the graph.
Now look at the graph that we are going to construct of the function 2 to the power of x. Of course I was going to put the number two there, right, you know that my favorite number is 2, right, oh well, okay. What do Gis do now to graph the exponential function, just like the others we always do.
Let's make a table here, draw up a table here, let's put values for X here to find the values of ydy but it's not f of x? Remember that I said that this notation f of x can also be written as Y = 2 to the power of x Okay, then remember that we talked about numerical value there I will put a value for x and find what the value of the function is here I calculate So , then I can take X being worth, if I take X being worth 10, which everyone likes to take We just talked here, an example will be 1, Ok, I can put a negative value of If I put x to be worth minus one, look what it will look like, it will be Y = 2 to the power of minus 1 to calculate 2 to the power of minus 1, remember what we did just now, I'm going to invert this base as a whole number. So I come here and put one and do the inversion which will be 1/2 to the power of 1, 1/2 to the power of 1 is 1/2 itself, so you can use the negative for you who are asking, can I use the decimal number?
But why would you want to complicate things with a decimal number? So, any number in the production statement, you can put any number in this table, here you can put any number. What did I say now?
Production isn't paying attention. It can be decimal, for example 0. 5, but then you will work with square roots, you will arrive at exact numbers.
So for us to complicate things if we can make things easier, that's not true, so let's always put numbers here that are integers, right guys, let's not put decimals now, let's for example put a positive one, if I put a positive one it will be 2 to the power of 1 which will be two itself if I put two here in place of good? If I put x as being worth two then I come here it will be 2 to the power of 2, 2 to the power of 2= 4 if I put x to be worth three it will be 2 to the power of 3, 2 to the power of 3= 8 and so on you can put other negatives If you want more, that's fine, that's it, that's it, and now what do I do, now you 're going to take a ruler, good ruler here in your hand, let's make our Cartesian plane to place these values that I found in the table, which are ordered pairs 0 and 1, - 1 and 0. 5, 1 and 2, 2 and 4, 3 and 8 and so on.
And just an observation: not at the time when production asked me here if these x values are in the statement, I said no, right, you create the one you want, but there are exercises when they are contextualized statements because here I'm talking about the graph of the function, but when you are solving exercises that are contextualized, it is possible that the statement will give you the value of We simulate the values, is it clear now? So let's do it. So, drawing my Cartesian plane, so here I plotted the values of y, one two three four five six seven eight, see that I plotted it very straight, cute, right, people to make the graph as best as possible for you to understand, okay?
Now here values for ordered pair so it is 0 of x and one of Y, so I mark it on the Y axis now there is a question that is recurring if it were 1 and 0 if it were 1 and 0 see that the first value is referent it is the xo second oy then there I would mark the 1 of x, ok if that were the case. So I'll delete this one soon. Negative one and a half or minus one is here the middle, right guys, it's right in the middle between 0 and 1 so I'm going to connect this ordered pair o - 1 and the middle then I'm going to make the ordered pair 1 and 2, 1 of X will connect with the 2 of Y, 2 and 4, 2 of , and then 3 and 8, look at 8 goes up there now let's delete my example here 3 and 8 and let's connect it to 3.
Here guys, it almost fell, that's it there are these points it's too precise I see that if you put the minus two for x if it were minus two place of X, how much would the value of our image be here? that would be one, and if I put two it would be a quarter, that is, it would be half of half, so it would be here, look at the little dot, it would be right here, so, as I said, it will never cross the X axis, it will lean towards zero as we said and So, when I have all these points, I'm going to connect them, look what's going to happen, so I know that it tends to zero, it passes through these points, wow, it's difficult to do here, look, here the behavior of ours here had to be corrected, okay? people?
This one here is the graph of our exponential function and see that here we have a graph that is increasing. Why is my graph increasing, did you pay attention because our a is a value greater than one, right? So every time this happens, oa being the value greater than one we will have an increasing graph and look where it crossed here o in this case in this type of function always crosses here at 1.
Here you can also observe that the domain that are the values assigned to X, it belongs to all real numbers so the domain of this function here we can say that it is all numbers real because it is any value that I can put for ox and the image the image of this function are those values of y, so see that here y belongs to real numbers, so that, now I will analyze, where is there an image where is there a X that will get your image there on the Y axis? When you see that it is hanging here, look, it will almost touch but it never crosses the than 0. So see that from here on out we can find values of x that look for its image on the y axis, okay?
Then domain and range of the function. Now I'm going to ask you when the value here is between 0 and 1, what will this graph look like? And now 1/3 to the x, how are we going to make the graph?
Pause the video and try to see if you can get it right. So let's go to our table if you want to do it straight away, x and y already. So let's put the values for x, I'm going to put - 2, -1, 0, 1 and 2, it's going to be symmetrical here, okay?
That one we messed up everything now in order. So if x is negative two, what will happen, people, then it will be of negative two, that's why it is the numerical value, right, so it will be 1/3, so instead of X, I put negative two when I have a negative exponent, I invert it. the base that will be three over one and three over one and three and the exponent will be positive two and three raised to 2.
9 so when I put the x being worth minus 2, the Y will be 9. When I put the x being worth minus one it will getting 1/3 to the power of minus 1 causes the inversion, it will be three when it inverts three over 1= 3 and three to the power of 1 which is three itself. It will be 3 here we already know that when I change it to zero every number raised to zero is one here when it is one it will be the base 1/3 itself and when it is two it will be a ninth.
And so on, how are you? You can now connect these ordered pairs in our graph, so let's go: minus 2 and 9 I'm going to connect minus 2 and 9 here. It will end up there - 2 and 9, with the capped pen you can't do it, minus two with 9 do straight here and that's it minus 2 and 9 then - 1 and 3, - 1 and 3.
I wonder if you can do this without a ruler, minus 1 and 3 that's it, I did it without a ruler. 0 and 1 so I mark 1 of y here, 0 and 1 and painted something with the pen 1 and 1/3, 1/3 if I divide to find out on the number line where it is I divide there, one by three dividing one by 3 I transform it into 10 tenths which will be 3, 3 x 3= 9 there will be one left so it will be 3333 so 033 is less than half a little bit so I'm going to do this approximate one and a third then it will be two with a ninth, which will be a little downwards as we already know our straight line, our straight line now was good, our graph never touches the graph drawing it never touches the X axis it always tends to zero there, it's OK? So now just call, let's go and call what's going to happen, now come upstairs and let me see if it works, okay , it's good, right?
So in this case, if you notice we have a value of a that is between 0 and 1. So our base a is between 0 and 1 and for this reason we have the graph that is decreasing look here look there you saw the other one as What did the other one do? As I was increasing the values of x, I was increasing the values of y.
Now here, I'm increasing the values of the one that increased in this one is the increase, this one decreases. So it is decreasing Ok? And then putting the domain, the domain will be all real numbers and the image of this function will be y belongs to the real numbers so that, and now as I look at the values of y, the values of y are the same as the other, right, it will always tends to zero so I never get here, so Y is greater than 0, is everything ok so far?
And one thing you can also observe is that when I graph the exponential function it does not go through the second, the second, the 3rd and the 4th quadrant here it is the first second third and fourth quadrant and then we have a long lesson on the exponential function right, and there was still a lack of resolution of contextualized exercises that I will do in another class for you to understand better and if you liked Gis's class on exponential function , leave a like for Gis, subscribe to the channel, share the class with your colleagues, give that help for Gis, that's summing it all up, and I'll see you in the next class, bye. . .
