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Contents

- 1 Goodbye Grant Hello Athena
- 2 Soda
- 3 Soda
- 4 Distance Driven
- 5 Distance Driven
- 6 Relationship Between Time Distance
- 7 Relationship Between Time Distance
- 8 Sodas and Money
- 9 Sodas and Money
- 10 Equations With Two Variables
- 11 Equations With Two Variables
- 12 Powerful Tools
- 13 Powerful Tools
- 14 Dependence
- 15 Dependence
- 16 Independent and Dependent Variables
- 17 Independent and Dependent Variables
- 18 Input Output
- 19 Input Output
- 20 Pairing Up
- 21 Pairing Up
- 22 Candy
- 23 Candy
- 24 Functions
- 25 Functions
- 26 Not a Function
- 27 Not a Function
- 28 What is a Function
- 29 What is a Function
- 30 Vertical Line Test
- 31 Vertical Line Test
- 32 Function Machines
- 33 Function Machines
- 34 Input x
- 35 Input x
- 36 Lets Code It
- 37 Lets Code It
- 38 DecreaseByThree
- 39 DecreaseByThree
- 40 f of x
- 41 f of x
- 42 Substituting Input
- 43 Substituting Input
- 44 Identifying Parts of Function
- 45 Identifying Parts of Function
- 46 Function Name and Independent Variable
- 47 Function Name and Independent Variable
- 48 What is f of 2
- 49 What is f of 2
- 50 What is f of m
- 51 What is f of m
- 52 What is f of negative m
- 53 What is f of negative m
- 54 What is f of x squared
- 55 What is f of x squared
- 56 Two Inputs
- 57 Two Inputs
- 58 Area of a Rectangle
- 59 Area of a Rectangle

Grant doesn't have time to work with us anymore, because he's so busy selling tons of glasses wipers and nozzles. And plus, he already learned a ton from us about algebra. So, he's referred me to someone else who he thinks is going to help put a new spin on ways that algebra shows up in the real world. Our new friend is his cousin, Athena. She says hello, and she is super excited to start working with us. Now, just to give you some basic info about Athena. She is a college student living in UdaCity. In school, she is majoring in architecture, but she's still taking classes in some other subjects, including college algebra. Athena loves hanging out with her friends and her family and she has a really cute dog and a really cute cat. So, all in all, Athena leads a pretty happy life.

One afternoon, as she was walking around in downtown Udacity, Athena happened upon a soda machine. And it looked so enticing that she decided to grab a drink. All the drinks from the soda machine cost $1.00. She put a dollar in the machine slot, pressed a button for which drink she wanted. And voila, out came a soda. She tasted her drink and since it was so delicious, she decided to buy one for her brother too, since she was about to stop by his house. So, she put another dollar in the machine and she got a second can of soda for him. Now, since the price per soda has the set value, a constant, of $1. What are two variable that we could very easily related to one another with an equation that uses this constant? Please pick two of these five choices down here. Would you want to use level of thirstiness, amount of money Athena puts in the machine, how hard Athena presses the button, how many brothers Athena has, or how many sodas the machine spits out? Which two of these belong together along with this constant of $1.00?

We could relate how much money Athena puts into the machine, to how many sodas it will give her.

As soon as she got her sodas, Athena hopped in her car and started to drive. Part way through the trip, her cell phone rang and she decided to answer the call. It was her brother, Nikos. He wanted to know how close she was getting to his house. So she started trying to figure out the distance she'd driven so far. Athena's been going at pretty much the same speed the entire time. So, what two quantities can she relate together in an equation that also involves that constant speed? Pick two of the four choices that are down here. Should the equation include the distance she's driven? The time her trip has taken? Her car's fuel efficiency? Or how much her car weighs?

Athena wants to figure out the distance she's driven. So that's definitely one of the quantities that we want to work with. And she can relate this to the time that her trip has taken so far, since we're using the speed.

Now that we know that we want to discuss the distance Athena has driven and the time she spent driving, which of the following statements makes the most sense in the context of her story. Do you think that time spent driving and distance driven are not related? Do you think that time spent driving depends on distance driven? Or, do you think that distance driven depends on time spent driving?

The distance that Athena has driven depends on the time that she spent driving. The more time that she spends moving at a constant speed, the further she will have gone.

Now thinking back to the soda machine part of our story, can we make a similar statement about the quantities we wanted to connect in that case? Which one of these choices do you think makes the most sense? Is it that the number of sodas the machine spits and the amount of money that Athena puts in are not related at all? Is it that the number of sodas the machine spits out depends on the amount of money that Athena puts in? Or is it that the amount of money Athena puts in depends on the number of sodas the machine spits out. Think about how cause and effect, or input and output play a role in this part of the story.

The number of sodas that the machine spits out depends on the amount of money that Athena puts in. In other words, the amount of money she puts in the machine determines how many sodas will come out of the machine. Without any money going in, there are no sodas coming out.

Since we've talked about several situations now that involve two different variables or two quantities that we might want to relate to one another in an equation. I think we should review what equations with two variables really mean. Now, I gave you this exact quiz way, way back in Lesson 5-1. But looking at it again might refresh our memories a bit. So, what can an equation with two variables do? Can it describe mathematical relationship between two variables? Can it show the output for a given input? Can it show how changing one quantity changes another quantity? Or can it demonstrate cause and effect? Please pick as many of these answers as you think are correct.

All of these options can correctly describe an option with variables, depending on the real life situation it's meant to represent.

So we can see from this quiz that equations with two variables are really powerful tools. However, not all variables that are connected, are connected in the same way. For example, some situations will involve cause and effect, but not all of them will. So bottom line, we need to think critically about the quantities we're dealing with, when we're forming equations, to figure out how we should connect the different variables that are involved. Remember that we can often represent quantities in the real world with variables and then connect these with equations. In these two situations that we just considered concerning Athena's afternoon, we saw that there was a relationship of dependence between the two things that we could express with variables. The number of sodas that came out of the machine depended on the amount of money that Athena put into it. And the distance she driven along the road depended on the amount of time she spent driving. In both of these cases the value of one variable helps determine the value of another variable. So one way we could describe what dependence means in the case of algebra at least, is that in an equation that involves dependence, the value of one of the variables helps determine the value of another variable. Now a couple of other situations that involve dependence between quantities surfaced during Athena's trip. Athena used the gas pedal while she was driving, and when she got to her brother's house she tossed him a can of soda. Now for each of these two new situations, please tell me which of the sentences below, probably describes the situation involving dependents.

The rate at which Athena's car accelerates depends on how hard she presses down on the gas pedal. Athena makes the choice about how hard to press down on the gas pedal and then that action determines how the car's motion changes. The second option down here doesn't really makes sense. The car doesn't change its own motion, and then affect Athena's foot. For the second one, the can's height above the ground depends on the amount of time that has passed since Athena tossed the can. The can's height doesn't control how time passes, time passes on its own. We can, however, look at how the height of the can changes from moment to moment.

Now in this course so far, we've seen a ton of different equations involving two variables. Things for example like y equals 3x plus 2. Now equations in two variables can express input and output, causing effect or some sort of other relationship involving dependence. The note seemed like the two variables in the equation would each play a different role. Perhaps it's that one of these two variables needs to depend on the other. Now just like in this equation right here, a lot of the other equations we've seen at different parts of the class, have been written in the same form. As y equals some sort of expression involving x. We saw that we could write quadratics in this form too. And there were a lot of situations in which isolating y by itself on one side was really conveinent for solving for different quantites. Now those equations were all written in the form y equals some expression where x is the variable. And if we have an equation that's written in this form, which of these three statements down here do you think is implied? In an equation like this does x depend on y? Does y depend on x? Or, is there no relationship of dependence between x and y? In other words, does one of these two quantities help determine the value of the other one?

When we write an equation like this, we're expressing y in terms of x. What we're doing on the right side of the equation, in this expression, is taking x, and then, manipulating the value of that variable in some way. That shows how it's related to y. If our equation is y equals 2x plus 3, for example. What we're saying is that, in order to get the value of y, take the value for x, multiply it by 2 and then add 3. And that will give you the number that you're looking for. Since the number we plug in for x is going to help determine the value of y, we can say that y depends on x.

We just said that for an equation written in this form, y usually depends on x. Now if this is true, how could we properly complete this sentence? Which of these two variables, x or y, is called the independent variable? And which of the two is called the dependent variable? Please write the choice you think is correct in either box.

In an equation like this, in order to calculate the value of y, we need to pick a value of x to plug in to the right side. And then that, in turn, will tell us what y is equal to. Now, of course you can always rearrange equations and solve for x instead, but we tend to write equations like this when we've assigned variables in such a way that y is generally what we're going to be looking for and x is more what we're given. So, for example in the situations we've talked about, the independent variable has been things like time, a sort of more fundamental quality, and the dependent variable has been things like number of sodas spit out, or height of a throw. Overall in the natural state of things, y is going to be determined by x.

Now, we've already talked about x as the independent variable, and y as the dependent variable, in equations that are written like this. So, we know we can think about the relationship between x and y, in terms of y depending on x or, alternatively, x being the cause and y being its effect. But we can also think about this in terms of input and output. In the soda machine example we looked at earlier, the input was the amount of money that Athena put in this sort of machine, and the output was the number of soda's that came out of the machine. So, for an equation like this, which quantity do you think represents the input, and which one do you think represents the output?

Since we need to plug in a value of x to end up with a value for y, x is the input and y is the output.

Now that we've talked about independent and dependent variables, and inputs and outputs, which of those are paired together? Please complete this sentence by filling input in one of these two blanks and output in one of these two blanks.

We can think of the independent variable as the input and the dependent variable as the output. So in an equation like y equals 3x plus 2, the input would be whatever number we plug in for x, and the output is the number that we get for y by using that value of x. The output depends on the input.

Now let's take another sort of toy example, of two quantities that we could very easily relate to one another, with an equation, where one of the quantities depends on the other quantity. And think about the relationship between those two variables a little bit more. So let's say that we have a candy machine, for which there is a set price for candy. So if I put one kind of coin in the candy machine, it gives me a certain amount of candy. So in this case the input, is the amount of money that you put into the candy machine. Just like for Athena, with her soda machine the input was the money that she put into the soda machine. And the output here, is the amount of candy that comes out of the machine, just like the number of sodas was the output for Athena earlier. So which of these two statements do you think makes the most sense? For a given amount of money put in, there's more than one possibility for the amount of candy that the candy will dispense. Or for a given amount of money put in, there is at most one possibility for the amount of candy that the machine will dispense. Think about how this machine works, and how you think that input and output should be related to each other.

For a given amount of money put in there is at most one possibility for the amount of candy that machine will dispense. This makes a lot of sense if you think about real life and how machines like this would work. If you put a dollar in the machine, a certain amount candy will come out. And the next time you put a dollar in, that same amount of candy will come out again since the price for the candy is set. If this first choice with the case, then any given amount of money could spit out a ton of different amounts of candy. It wouldn't be a very fair machine. And people would probably get really frustrated and not buy candy from it anymore.

Now the relationship between the variables in the candy machine situation we just talked about is of a special type, it's called a function. Now some more strictly mathematical examples of functions are things like I've written right here. Y equals negative 2 x plus 5. Y equals x squared plus 5 x plus 6. Y equals negative 5. There are ton of other ones. Pretty much all of the equations for lines and problems that you've seen in this course are functions. We just haven't been using this word yet. Now, however it is time. Think back to the relationship between the input and output in the gumball examples from the last quiz. And then please complete this sentence here for me. If we consider x to be the input and y to be in the output of some function. Then there is no more than one value of, what for a given value of what. Remember again, how you answered the similar question for the candy example. You can also, of course, look at values of x and y that satisfy each of these equations, or you can graph them to figure this out.

So, for any given function when we're considering x to be the input and y to be the output, there's never going to be more than one value of y for any value of x that we look at. Now, since I told you that these were some examples of functions, why don't we take a look at each of them and see how this concept applies. So first, you have the line y equals negative 2x plus 5, which I've drawn for you down here. And we can see that as we move across the horizontal axis, we're never going to be able to move up to more than one y value or down depending on your position. Now, for this parabola. Once again, even though this isn't a straight line, if we move horizontally across the graph, we'll never encounter more than one vertical coordinate for a given horizontal coordinate. Y equals negative 5 is just a horizontal line. And we can see, once again, the same concept applying.

Now, that you've had a definition of what a function is, let's look at some graphs of different equations and see if we can tell which of these are functions. So, remember that we defined a function as a relation between variables, where there's, at most, one output for a given input. And for all of these graphs, x is the input and y is the output. So, if we can't have more than one value of y for any given value of x, for any curve that represents a function, then which of these 6 graphs over here is not a function?

And the answer is A, this vertical line right here. Since every single point along a vertical line has the same x-coordinate, but a different y-coordinate, this definitely doesn't fit the criterion for a function that we've put up. However, for every other curve or line on this graph, each horizontal position only has one corresponding vertical position. Even though, two of the graphs on here, F and D, don't represent equations that you've seen before, you can still tell, right off the bat, that both of these are functions. Even though you don't know anything else about them.

Thinking about the conclusions we drew about these graphs in terms of which ones are functions and which one isn't, let's make a generalization about how we can tell graphically what is a function. So in other words, how can we test graphically that a given curve only has 1 y value for every x value? I'd like you to complete this sentence to create this guideline for us. So if we draw what kind of line on the same coordinate plane as any given function, the functions curve should intersect that line some number or fewer times. Now this is definitely a really tricky question, seriously. So if you have trouble with this that is 100% fine. This is going to give us a really useful tool. So take a second to consider the question.

If you draw a vertical line on the same coordinate plan as a function, and the functions curve should only intersect that line 1 or fewer times. Let's pretend for a second that this pencil is a vertical line. Now saying that a functions curve should intersect this vertical line 1 or fewer times, is exactly the same as saying that each x value should have only 1 y value associated with it. Remember, just like line A over here, and like this pencil now any vertical line has an equation that can be written in the form, X equals some number. 4 in the case of line A right here. This means that every single point along this line will have the same x coordinate. It also means that if another curve intersects a vertical line at more than spot, and there's more than one point on that curve, that has the x coordinate of the vertical line. So even if we drew something like this. A little bit messy but you get the picture. This blue curve is not a function. Why is it not a function? Because if we move our pencil to, actually a tiny different positions along the x axis, this curve would cross to the pencil in more than one spot. That means that for a given input, a given x coordinate, you might either end up with one y coordinate, or a different y coordinate. So vertical lines and other functions that don't pass this test over here, don't work the same way that the candy machine does. If we had a candy machine that worked like this, for multiple values of amount of money you put in, you could end up with different amounts of candy.

The rule that we just came up with in the last quiz for figuring out graphically whether or not a curve is a function is called the vertical line test. So remember, according to the vertical line test, we can tell that a curve is a function if there is no vertical line that we can draw anywhere on the coordinate plane that intersects that curve at more than one point. So, using your newfound understanding of the vertical line test, which of the five curves over here are functions? So, please check off as many as you think are, in fact, functions.

The answer is that all of these curves over here are functions, except for curve D. That's this light blue one right here. And we can see, that right here between 0 and looks like about 4, if we put a vertical line in that region the curve that intersect that line at more than one point. That means there are certain inputs, or certain x values, on this curve for which there is more than one y value, so this is not a function. All the other ones are however, including another graph unlike anything you've seen, this graph E right here, this is just proof that there is always more to learn in math.

So, now you've seen graphical depictions of functions. We've talked about real world examples of functions. And I think it's finally time to talk about what a function really is mathematically. So, overall what does a function do? Well, it takes some value as it's input, does some stuff to that value, and then produces a new number called the output. Then we can pair that input value with the output value that it produces to graph our functions. So, over here we have our x's and over here we have our y's, in general. So, for example, let's say I have a function whose function is to increase the input value by 2. So, for example, if our input is 2, so if you take 2 and throw it into our function machine, then the function machine will spit out an output of 4. So instead of 2, our input is

the manipulation letter function here performs on the input value, is to increase it by two. So if our input is five than our output is 7.

Now, what if I want to be a little bit more general here? What would the output of this same function, the one that increases things by 2 be, if I give it an input of any number? Let's call that number or that variable, x. So if x is the input, what should the output be?

Well, if I input some number x, I want my function to increase this by 2. Which would mean that the output would be x plus 2.

We just said that if we want to make a general statement about what the output, based on a general input for this function we're talking about here, would be, if the input was x, then the output would have to be x plus 2. Since the function of our function is to increase all input values by 2. So, to illustrate this point a little bit differently, we're going to so some very, very basic computer programming. Now, don't worry if you've never done anything like this before. This is just to illustrate how simple function works. You'll never be tested on this or anything. This is just for you to explore and have fun and maybe even understand things a little better. Now what we're going to do is to use one particular programming language called Python, to create the same increase by 2 function that we just did. Now remember, that we said in a general case of wanting to increase by 2, we take an input of x and then have our output be x plus 2. It's the same thing as happening up here in these two lines of code. The letters def stand for define and they're saying that we're going to define or create a function, which is named IncreaseByTwo, and that function is going to take in some input, this letter x in this case as the general case. And that function is going to do some stuff and then spit out this output, x plus 2. So, these two lines of code are basically creating the machine that this function is. The machine that transforms whatever input value it gets into some output value. Now, please notice here that we do have parentheses here around the input value. This is absolutely nothing to do with multiplication. These parenthesis just indicate, this is what we are calling the input. Lets try it out. Lets see if this function works. Lets say that I want to use an input of 7 or plug in 7 in place of x in my IncreaseByTwo function. What number, do you think, this program is going to spit out in the end. Now again, you don't need to know anything about how programs actually work. All you need to do is think about the relationship between the input here and the output here. And see, then, what the output would be if 7 is the input. Please put your answer in this box over here.

Well, I've typed this in and all I need to do is hit enter to find out what the answer is. And we get a value of 9. This makes perfect sense. You know that 7 plus 2 is 9. Now, we can try this with other values too. Let's play around. I could do increase by 2 of 25, let's say. That gives us 27. I could do increase by 2 of negative 17. This time let's make an interesting number like 0.9886. That gives us 2.9886. Awesome. So, hopefully this is giving you another way at how functions work. We have three clearly defined things involved in this program right here. We have the input. The function, which is processing this input. And then the output that the function spits out.

So now you've seen what kind of function we write using Python to increase the value of any input number by 2. In other words to make the output value be 2 grater that the input value. Now maybe you can create your own function. Can you design a function that will take some sort of input and then of the output, be 3 lower than that. Let's see, a logical name for this function then would be, decrease by 3, and once again I'm going to call the input x. I'll type in the return for you. What I'd like you to do is to fill in what you think belongs right here. What general expression representing the output belongs here if our input is x and we want our function to decrease the input by 3.

Well, if we want our function to decrease things by 3 then we should take our input x and subtract 3 form it. So our return value or our output should be x minus 3. Let's play around. If I say, I want to figure out decrease by 3 And have the input be, four, then sure enough I get one. Let's do a couple more. We can see that in both of these cases, the number that is output is 3 lower than the number that is input. Which is exactly what we wanted to have happen.

We just saw that in computer science, you might write a function like this. First we name the function, in this case we named our function decrease by 3. Then we create a spot for the input, which we call x and put it in parentheses, and then we define what the output is going to be with this return statement. And then we write the output right here, x minus 3. But as you've probably come to realize over the course of this class, some people who do math a lot are a little bit lazier, like me. For one thing, in math, the name of the function is not usually nearly as inventive as it is in computer science. We can see that a programmer would call a function something like this, decrease by three, that really describes what the function's doing. In math though, our names for functions are really simple. In fact, the most common one is probably just the letter f. What programmers and mathematicians do in common however, is that they still use the parentheses around what's going to be the input for the function. And of course, x does seem like a pretty great input. Now the programmer has a return statement, that represents what the output of this function's going to be, only around the program. But in math, we just create an equation. So if we want to create the decrease by three equation in math instead, we just write f of x equals x minus 3. The output is just that expression that we've often seen written to describe the deep ended variable, showing how it depends on the independent variable. Now the way that we read the left hand side of this equation right here is we say f of x. Now again, just like in programming, the parentheses here around the input do not mean multiplication. So the output of the function, the thing that's just equal to x minus 3, is this f of x over here. And remember we used to say that the output was y, if the input was x. So, what we're really doing in functions is replacing the independent variable, y, with this new way of writing it, f of x. I think you'll see over time why this is a useful way of writing it. But it doesn't make the original way we learned any less valid. They're just different ways of expressing pretty similar things, assuming that this is a function. Now that you've seen how both computer science and mathematics would represent the same function in different ways, I'd like you to translate the first function that we talked about from Python into its mathematical form. So this is the function that we initially called Increase By Two. I'd like the input to be x, and the output to be x plus two. I'd also like you to name the function f. So just like we did up here, write the mathematical form of the function down in this red box.

This time, our function is f of x equals x plus 2. So you can see that the input is still x and the output is still x plus 2. The only thing that's different is that we've chosen, once again, a super simple name for our function. Just the letter f.

So, now that we've seen a couple of different functions, let's talk about how to actually use them. When we had a computer program, we saw that we would take our function, type the name out, and then in the parentheses, where we know that the input goes, we could type a specific number like 4 for example, like we did before. And then because the function was already defined in the program, when we ran this, with an actual number as the input, it would give us an answer. It would spit out in this case a 1. We can do a similar thing with the mathematical version of this function. Once again, in the slot where the input goes. Inside the parenthesis right here, we can write a number. Let's pick a different one this time just to spice things up a little bit. Maybe 5. Now the way that we use this new input of 5, is we substitute it in to all the places where we see x in the expression on the right hand side over here. So, in fact the first term you see is an x, so we replace that with the 5. And then the rest of the expression is the same. We started off with f of x equals x minus 3 and that means we know that f of 5 is equal to 5 minus 3. We can evaluate this to see that f of 5 is equal to 2. So 5 here is our input, and when plugged in to this function, the function relates the input of 5 to an output, which in this case is 2. This in fact creates a point for us. We can now plot the point 5 comma 2. This is a point that lies on this line. Since we know that we're just replacing our dependent variable with f of x instead of y, we graph all the values of f of x of the function of the output just as we would y values on the vertical axis. So, what if I have a different function instead? Maybe f of x equals x squared. What if I want to plug in 4 as the input for this equation? What would the function give me as the output then? Please simplify your answer as much as possible, so that it's just a single number.

As before, we're just going to substitute four in for x in the original function. So we're going to end up with 4 squared with is just equal to 16. So our input of four spits out an output of 16. And once again this creates a point for us that we can plot a 4 comma 16. This lies on the graph of f of x equals x squared. Which incidentally is a problem you know very well. So the way that the graphs of these functions look hasn't changed at all. From when we used to write them with y's instead of f of x. So the graph of y equals x squared is exactly the same as the graph of f of x equals x squared. Writing with this notation just underscores the fact that this is a function. This has that special property of having only one y value for every x value that we can plug in. And it also shows that the vertical values on this graph are dependent, upon the horizontal values.

I've given you several functions written out here. And for each of these functions, I'd like you to identify the name of the function, the input of the function, and the output of the function, in terms of the input value. So I've already filled in how you would do this for this first function. Which we would read as h of t, equals negative 9.8t squared. So here you can see that the name of the function is h, the input is t, and the output is this expression over here. Now you have 9.8t squared. So please fill in all of these boxes for the other 2 functions.

For the first of these, a of F equals F over 10. The name of the function is a, the input is the letter capital F, and the output is F over 10. This function may look a little bit funny because we haven't seen anything in this form really, or with these letters as variables. But this function does describe something really important. This describes how the acceleration of an object depends on the force that's been exerted on the object. Now let's look at the last function down here. The name of the function is p, the input is the variable c, and the output is negative 10c squared plus 100c. This looks pretty reminiscent of the one of the equations we used with Grant where profit depended on cost, or rather, the price that he charged customers.

Now you've seen that mathematicians like to name functions simple things, like the letter f. And we've also seen that the input of a function is some sort of independent variable, and we've seen that x can be that variable. But we don't have to use these names, f and x, for these functions and independent variables. We can actually use pretty much anything we want to use. Remember, variables are just placeholders for things that have different meanings in different real world situations. What if instead of f of x, we have g of u? In this case, what is the name of the function? And what is the independent variable that's being used in that function?

The function name is g. This letter out here. And the independent variable is u. The letter found inside of the parenthesis.

So now, that we've got an idea of how to use function notation in the context of equations, how do we actually use it? Well, it gives us a sort of short hand, as we saw in a couple of quizzes before. For instance, if someone walks up to me and asks me, what is the value of x squared minus x plus 2, when x is equal to squared minus x plus 2 is a function where x is the independent variable, the same thing is asking, if f of x equals x squared minus x plus 2, then what is f of 2? So actually, just because I know you want to solve this problem, what is f of 2? Please fill your answer into this box.

All we have to do, as we saw in the simpler examples before, is take the original function, and then, substitute 2 in a spot of every x. So we get f of 2 is equal to 2 squared minus 2 plus 2. Which is just equal to 4.

Although so far we've only substituted in numbers in place of our independent variables, we can really substitute in pretty much anything we want. Now, you know that my favorite letter in the entire world is m. So, I'd like you this time to set x equal to m, and use that in our function. What is f of m?

Once again, we substitute m in place of x. And we end up with this expression, m squared minus m plus 2. We can't simplify this any further, since none of these are like terms. So, this is our final answer.

We just substituted in x equals m, but, what if instead, we have x equals negative m? What is f of negative m? Please write your final answer. Simplify it as much as you can in this box.

So each x that we see we want to substitute negative m directly into that spot. Since we know that the entire quantity x is squared, we need to make sure that the entire quantity negative m is squared. So I put parentheses around it to indicate this. The same thing is true with subtracting here. You have minus negative m and then plus 2. So let's simplify this, negative m squared turns out to be just m squared. Since the negative signs will multiply with one another and turn in to a positive sign. Minus negative m is the same as plus m and then we have plus 2 at the end. Great!

Now, I think this is a really difficult concept, so fair warning, this might be confusing. If you're totally good on this, awesome, move forward. If not, you might want to spend a little bit of extra time trying to understand this concept. Since we said that we can substitute really whatever we want in the place of x, I can actually substitute in a different expression that involves an x. So here we have a function defined with x as the independent variable. And in the place of a independent variable I can plug in x squared for example. Remember that these variables are really just place holders for us to plug in, other numbers that we know that they can equal. And if we decide that we want to plug in x squared, we are totally entitled to do that. So, how would you write out, in simplest terms, what f of x squared is? And yes, please remember to simplify as much as you can.

As before we need to make sure we apply whatever operation is acting on x to the entire quantity x squared. So I start by putting that in parentheses and then raising the x squared to the 2nd power again. Then we subtract x squared and then we keep that added 2. This gives us a final answer of x to the 4th minus x squared plus 2.

So you've seen functions written in the form f of x or g of u, or any other number of things like this, h of y, so on and so forth. You can pick any letter you want for the function name, and pretty much any letter you want for the independent variable name too, as long as you're consistent. So taking all of that into account, what do you think this means? F of u comma v. Does this represent two separate functions? Two outputs for one input? A function with two independent variables? Or a function with two dependent variables?

This represents a function with two independent variables, u and v. And this is probably a little bit hard to grasp, so I'll give you an example in just a second.

We just talked about variables written in the form like this, f of u comma v. So, we have two independent variables and one dependent variable. An example of a situation like this in real life is finding the area of a rectangle. Note that the rectangle area is going to depend both on the length of the rectangle and the width of the rectangle. So, if we choose to call the length, l, and we choose to call the width, w, and we choose to name the function A, how can we express the area of a rectangle as a function of length and width? Please write this in form f of u comma v.

We can express the area of a rectangle as a function of length and width by writing it as A of l comma w. And we know that this would equal l times w, since we find area by multiplying length by width.