Welcome, this is Dr. G, and in this video I will be covering five extremely important math concepts that you should know before you start calculus. Of course, that's not to say that these are the only five things you should know before going to calculus. Ideally, you would know everything for pre-calculus before going to calculus, but that's not possible and that's just pointless.
What I'm trying to say is these five things that I'm going to cover with you are actually very easy, but they pay off big time and they make your life so much easier in calculus. You got this. Come on, let's have some fun.
Let's go. First of the five is function notation. And in my opinion, it's the most important to Let's start with this function here.
x is known as the input or the independent variable. We get to select whatever number we want and replace x with that number, and it will undergo some type of algorithm here. As you can see, it will get multiplied by the 2 and then added to a 1, and whatever answer we get after that is known as the output. Well, since y is equal to that entire right side, that means y has to equal to the output as well, so y is known as the dependent variable or the output. That is the basic premise of how a function works.
As you probably know, know we can rewrite the y as f of x. That means f of x is also going to be the output. Nothing else changes here. x is still the input, and now notice x is also on the left side in a bracket. What does the f mean?
Well, the f is just a name of the function, and it doesn't have to be f. It could be any letter in the alphabet. f of x really means function of x, which actually means the result or the output if you were to choose x as your input. So if I had f in the brackets, that means it will be the result of the output if you were to choose 2 as your input instead of x.
f we said is the same as y, and since y is the output, f is also the output, and we haven't changed a single thing on the right-hand side, so that remains an output as well. Let's take a look at the function on the left first. As you can see, it's clearly a linear function, something that you've probably learned before if you're taking calculus.
In a linear function, the equation for this one is y equals 2x plus 1. Now we can also write it as f of x equals 2x plus 1. Cool. What about the function on the right? It's going to be pretty hard to represent this as one equation because there is no clear-cut pattern.
This is called a piecewise function and we're not going to get into how to actually figure out the equation for this function, but what I want to tell you is if you see a function that you don't know the name of or that you don't know how to represent as an equation, you can always just call it y equals f of x. Even if you know the name of the function, you know the name of the functions, look on the left, we know this is a linear function, we know the equation, there's nothing wrong with calling this y equals f of x as well. Now what if they were to ask you to find f of 2? That means they want to know the output or the result if your input is 2. So we're going to just write down the equation equals 2, but where the x used to be, because that's the default input, now we know the specific input is 2, so we'll replace it in brackets with our 2 plus. plus 1. And then to solve this, that becomes a 5. So the output, when your input is 2, is 5. Your inputs don't even have to be just numbers.
It could be variables and numbers, like the next one, x plus 2. Nothing changes here. All we're going to do is we're going to replace our default input of x up here in brackets with our new specific input of x plus 2. And don't forget the plus 1. And then we simplify it, and that becomes 2x plus 4 plus 1. And that gives us... 2x plus 5. Now the third example is the opposite.
We're looking for x, or the input, when we know the f of x, or the output, is 7. So I'm not going to replace the x with my 7 here, because 7 is the output. I'm going to replace the entire f of x with 7. So the equation becomes 7 is equal to 2x plus 1. Notice I left the x as it is, because we don't know what x is. We don't know the input. And then we just do a quick culture better solve.
Subtract. one to get six equals two x and then we ultimately get x equals three as our answer. This one might look weird because our specific input that they want which is x plus h has no numbers in it but that's fine we're going to just stick with what we know we're going to replace our default inputs of x with x plus h so it should look something like this and then with a little bit of simplification foiling expanding the brackets it will look like this. Another application of function notation is to represent coordinates.
of a point on a function. So let's take this function. We can say that there will be a random point with a coordinates x comma y that's located on this function.
Absolutely. But didn't we say that both y and f of x are outputs and they're the same? So we can also say that the coordinates of this point is x comma f of x.
Notice we replaced the y value of the coordinate with f of x. Let's take it one step further. Did we not have on the board here f of x is equal to 2x plus 1?
Well if they're equal then I can replace the f of x with 2x plus 1 and that's also a way to represent the y coordinate of our point. So as you can see, these three ways here are different ways to represent any random point on our function. You also need to know how to find inputs or outputs graphically, not just algebraically by plugging in.
Let's take a look at this graph and look at the first question. They're asking for f of negative 4, meaning when the input or the x value is negative 4, what is the y value or what is the output? So just scan the graph.
Look for where the x value is negative 4, right here, and where is that point? Well, the y value of that point is 0, so the output is 0. Same thing, f of 6. When x is 6, what's the y value? It's 4. f of 2. Well, when x is 2, notice that there is a circle above it and a dot below it.
Remember, an open circle in math represents non-inclusive, so there's actually no point there. It's hollow. But luckily there is a point down here, so when the input or the x value is 2, the output is negative 4 for your y value.
Next, f of negative 2. When your input is negative 2, notice there's two empty circles here. There's nothing there, but luckily up here there is a random dot where x is negative 2 and the y value is 6. We'll take that. The y value or the output is 6. Next, f of negative 6. This time. Do you see anything along the line of x equals negative 6, aside from this empty circle?
No, which means there's no point, there's no y values at all when your x value is negative 6, so we put dNE, or does not exist. Lastly, they're asking for x when your f of x is negative 4. That means this time they're giving you the y value, or the output is negative 4, they're asking for the x, or input value. So let's scan up and down this time for a y value of negative 4. As you can see, it's down here, and there is a open circle to the left, very useless, but luckily there is a closed circle to the right, and the x value is 2, so the answer is 2. Lastly, let's see how we can represent some of these as coordinates of a point. So the very first question here, f of negative 4, we know that the x value is negative 4, and we just figured out the y value is 0. That means this point right here is negative 4. should have the coordinates negative four comma zero.
Yes, but we can do this a different way as well. Our input or x value is negative four. Okay, then our y value, we can represent it as f of x, remember?
But not f of x because our input's no longer the default x. Our input is specifically this time negative four. So the point can be represented as negative four comma f of negative four. The second important concept is linear functions.
The good news is you don't have to know the entire chapter or everything about linear functions. There's only two main things you got to refresh your mind of when it comes to linear functions before you start calculus. The first thing we're going to talk about in linear functions is slope.
Well, slope is represented by the letter m and it's calculated with rise over run, but a much more useful equation to calculate slope is y2 minus y1 over x2 minus x1 given that you know two separate points on the graph denoted as x1 and y1 for your first point and x2 and y2 for your second point. Let's take a look at this function on the screen. Some of you might be thinking this is a quadratic function or a parabola and you're probably right but we're going to be lazy here and we're going to call this y equals f of x.
Let's find a point on this graph shall we? Let's say this point right there. Okay what is its x coordinate? We don't know do we? So we're going to call the x coordinate x.
If the x coordinate is x, then what about the y coordinate? Would the y coordinate be y? Sure, but you'll see later in calculus, we don't like that.
So what's another way we can represent the y coordinate? Remember earlier, we can represent it as f of x. So the actual coordinates of this unknown point can be written as x comma f of x.
Okay, let's do another point. Let's say up there. What is the x value of that point?
I just hypothetically said x plus 2 because I have no idea what it is. So if the x value is x plus 2, then what should the y value be? Ah, x plus 2 comma f of x plus 2. So all of a sudden, now we have the full coordinates, although they're not actual numbers, but whatever.
We have coordinates for both points, which means can we now find the slope? Of course we can. If we have two points, we can plug it into our slope equation and find the slope.
But let me ask you this. What slope are we actually finding if we're using these two points? Because a constant slope only works for linear functions.
This function is not linear, which means if we were to find a slope using these two points, we're actually finding the slope of an imaginary linear line that connects these two points, and that line is called the secant line. We're not going to go too much into this, you will learn this in calculus, but for now, just know that when you're finding two points on a curved graph, that slope you're finding is not the slope of the curvature. The slope is just that of an imaginary linear line that connects the two points. Let's put these points into the slope equation.
Be careful to label what's your x1, what's your y1, and all that. And once you plug it in, it should look something like this. We can simplify the bottom. The x is actually cancelled out, so our slope will look something like this. I know it's not a real number.
It's not an actual slope number. But as long as you know how to do this and why you're doing this, you should be fine for calculus. The second thing you got to know for linear functions is to how to take the slope, which is something you will find, as well as a point, something that I'll probably give you, and combine them into the point-slope form, which should look familiar.
And then from there, you could turn it into the slope-intercept form, y equals mx plus b, where the b is the y-intercept, but it's not that important here. Some teachers are actually okay if you leave your answer in the point-slope form, so there's no need to go into y equals mx plus b. You can also get it to the general or standard form, but honestly in calculus you will never need to do that, so don't worry about that form. The third concept you must be good with before starting calculus is exponents.
Here, I'm stressing on three specific rules or applications of exponents. Let's take a look. The first one is rational exponents.
For example, if you were to have something like 3 to the power of 3 over 4, that can be written as the fourth root. of 3 to the power of 3, or you can write it as the fourth root of 3 and then all to the power of 3. We can also go backwards. Maybe we have something like the square root of x to the power of 3. We can write that as x to the power of 3 over 2. Going this way is way more important than the other way we just did. In calculus, you have to be very very comfortable going from a radical to a rational exponent.
The second rule is negative exponent rule. Up until now, you've probably used it mostly to get rid of negative exponents and turn them into positive exponents. Well, in calculus, that might happen, but for the most part, you have to take something with a positive exponent and rewrite it as a negative exponent just for the sake of getting rid of fractions.
The last one's a bit funky here. In this case, you have to be good at taking out the greatest common factors with exponents. Let me explain. Let's take a look at this expression here. Can we find a greatest common factor?
And the answer is yes. They all share x to the power of 1. Well, how do we factor out the greatest common factor? We divide all terms by x to the power of 1, or just x.
And when you do that, we're using quotient rule. So we're subtracting. So you end up with x on the outside, because that's my greatest common factor.
And on the inside, x cubed over x is just going to be x squared plus 2x minus 4. That's the GCF. In this case, let's say they force you, or you're forced, to factor out an x to the power of 4. Well, you might be saying, not every term here has x to the power of 4. Only the first two terms does, and then afterwards, they don't have enough x's. It doesn't matter.
You still have to factor out the x to the power of 4, and it's possible if you just stick with the quotient rule. So let's see here. We're going to divide each term by x to the power of 4. And when we do that, we're going to subtract the exponents, and let's see what happens. The first term becomes 2x, the second term becomes 4, and then starting with the third term, we actually have more x's on the bottom than the top, so it becomes 8 over x, plus, and then here we have two more x's on the bottom, so it'll be 1 over x squared minus 1 over x cubed, plus, and then we keep the last one exactly the same, 9 over x to the power of 4. That is what you have to do here.
It might look weird, it might feel awkward, but you need to know how to do this for calculus. The fourth concept is domain and range. Recall there's two different notations for writing a domain and range.
There's interval notation and set notation. In the interval notation, the brackets mean everything. A round or open bracket means that it's not inclusive, whereas a square bracket means it's inclusive. Non-inclusive on a graph would be something that is like it's open circle, that means we don't have an actual point there, whereas inclusive means it's either the graph running right through or there's an actual filled in dot on the graph. In interval notation, we also use the infinity sign for positive infinity, as well as negative infinity if we're going towards the left.
So a typical domain or range in the interval notation could look something like an open bracket with a negative infinity, for example, and then it goes to like four, and let's make it inclusive. Keep in mind, if you're going to use infinity or negative infinity, you can't ever include infinity, because infinity literally means it keeps going on and on. so it should always be paired with an open bracket when it comes to infinity now for set notation that's something where you would use stuff like greater than or less than or greater than or equal to or less than or equal to and you would usually set it up like with a bracket here put x if you're doing domain with a vertical line then you can say like x has to be greater than or equal to 4 comma xcr and then close the bracket some teachers don't even care about the brackets you can just write x has to be less than three or xer if you're doing domain and then just replace the x's with y's if you're doing the range that's set notation but unfortunately in calculus you don't actually need to use too much of this so focusing on the interval notation you're going to need to use this when you're doing something called curve sketching and they're going to ask you at which intervals is the graph increasing or decreasing or concave up concave down so just know how to use interval notations and then there's this thing here there is a u And this stands for union. Now, before calculus, if you were to have two separate domains or two ranges, you would do something like this. You would say it spans from, let's say, negative four to zero inclusive.
And then you can write the word and or you can write or. It starts from positive two and it goes to positive ten inclusive, something like that. But instead of the word or or and.
you just replace that with the u so it will look like negative four comma zero and then u for union and then two comma ten just keep in mind that's what the u stands for it just means and or or last but definitely not least is the concept of composite functions a composite function is when you take the outputs of one function and you substitute it into the input of another function simply put you're just gonna take one entire function and shove it into where the x is in the other function. Let's take a look at our two functions here. We have an f of x function and we have a g of x function.
If they're asking for f brackets g of x close brackets, notice the x used to be in the brackets for the f of x. It's not there anymore, is it? The x or the input got replaced by the entirety of the function.
of g of x, which is the output of your g of x function, isn't it? So we're taking the output of g of x, or the entirety of g of x, and we're placing it where the x used to be in our f of x function. How do we actually do this?
Well, we're going to first write the f of x function, but whenever you see an x in the f of x, that has to go now. That's got to be replaced by g of x, so there you go. We do that to both the left and the right. Now on the right side, recall that g of x is actually x squared minus 3x plus 10. That's the actual function output of g of x. So we can substitute that on the right.
You don't have to do that to the left. The left is just there to show that we're doing f of g of x. Now on the right, you can continue to simplify and simplify until you get this composite function.
One last note, this composite function, what they're asking for, can also be written in a different way, like this. f of g x, the o is actually of, so it's f of g of x. That just means the second letter, or the g, is getting stuffed into the first letter, or the f.
Okay, so now you know what a composite function is, but unfortunately in calculus, you actually have to look at a composite function and be able to pull apart the f of x and the g of x. We have a bracket with x squared minus 4x inside, all to the power of 3. Well, there's actually two functions in here. We can think of it this way.
on the outside, the brackets to the power of 3, could there have been just an original x in there? And then we stuffed the x squared minus 4x into where the x used to be, so now it's inside the brackets. So in this case, we have f of x, which is just x cubed. I'm putting brackets around the x to show that the brackets are there.
And then the g of x, which is what we're stuffing into the f of x, the g of x is x squared minus 4x. How do you know you're doing this right? Well, let's go backwards, right? If we were to stuff this x squared minus 4x, into where the x is in the f of x, you would get that. Okay, let's try another one.
Log 2x cubed. Well, in this case, what stands out right away? It's the log, isn't it? Okay, well think, what's the default function for a log? Well, the default function for logs is just log x.
But I don't see log x, I see log 2x cubed. That means the 2x cubed is probably stuffed into where my x used to be. So my f of x here should be my default log of x.
Again, I'm putting brackets that show the stuffing here. And then the g of x is the 2x cubed. And as you can see, if that goes into where the x is, we're going to end up with that. Now, the problem with this one is it's in a fraction. So the very first thing we have to do, we don't have a choice here, is to move the sine 2x to the top.
Although we get a negative exponent out of it, that's okay. Not a big deal. Now, looking at this, does this look like a...
Composite function. Yes it does. Because why? We have a bracket all to the power of negative 1. That's got to be my outer function.
That's got to be my f of x. Well what about the inside? The inside is sine 2x.
Well hold on a second here. The sine stands out. And what's the default function for a sine? The default function is just sine x, isn't it? But I don't see that.
I see sine 2x. That means we have another function stuffed inside of my sine function, which is stuffed inside of my brackets to the power of negative 1. So as you can see, we have a composite and a composite. The 2x is stuffed inside of my sine x, which is stuffed inside of my x to the power of negative 1. Alright, last example of this video here.
Starting off with the problems here. What do we see? Well, we have a radical on the bottom. We don't like radicals in calculus. We like rational exponents.
So let's write this as 2 over brackets x squared plus 2x to the power of half. Okay, what's another problem? Another problem is we have a fraction. We don't like fractions in calculus either, so we're going to move this bottom to the top, making it a negative exponent, but that's fine. So let's move it to the top and we're going to get 2 bracket x squared plus 2x to the power of a half.
Okay, and at this point we can see that we have a outer function that is 2 and then a bracket to the power of half, so that can be my f of x here. My f of x can be 2, and then that bracket should just have an x inside, keep it simple, to the power of a half, which means the stuff in the brackets that replaced the original x has to be my g of x. So the g of x should be x squared plus 2x. All right, we're done the video, finally. For those that made it to the end, awesome job, and I can promise you it will pay off when you start calculus.
Like I said, these five things we covered, they're not that difficult. They're actually pretty basic, but if you know them really well, it will make your life in calculus so much easier. If you like the video, remember, give the video a like. And if you really like the video, subscribe to my channel, help it grow. And as always, if you have any questions or comments or any topic requests, leave it in the comment section down below.