Athena started interning at an architecture firm, to learn the basics of designing and planning for the construction of new buildings. This isn't just any architecture firm, though. It just so happens to be the firm that designed the new office for Grant's gleaming glasses company. So we're going to go back in time to follow Athena through the work that went into creating Grant's awesome new space. Since Athena is just getting used to being a part of the world of architecture at all, the firm gave her a pretty simple project to work on first. Since Grant's such a fun guy, he wanted his office to have a separate space, just for having fun. Thus, the fun room, which Athena get's to plan. Grant has picked out a very special wallpaper that he wants used in this room. And has a pattern that he wants to make sure is fully expressed on every wall, but he also wants there to be natural light in the room. So he wants windows along two of the walls. However, since he's not willing to sacrifice any of that wallpaper pattern, the two walls with the windows on them are going to have to be longer than the other two walls. Let's call the distance from one wall to the edge of a window x. And assume that each of these windows are placed in the center of either wall. So that there's a distance of x on either side. Now, since Grant loves natural light so much, he wants these windows to be really big. In fact, he's already picked out, surprise, surprise, the exact windows that he wants. And they are each 3 meters across. Now, since we're calling each of these lengths of wall x, we know that this length of wall is going to be equal to the sum of those two. So this is just 2x. Now, the thing that Athena needs to figure out is how much carpet is going to need to be laid down in the fun room. So considering these side links and your understanding of, area of rectangles, please come up with an expression for the number of square meters of carpet Athena's going to need for this room. You could either multiply or not multiply out your answer.
We saw at the end of the last lesson that we could come up with a function for the area of a rectangle by saying that A of l,w equals lw. The product of the length and width of any rectangle. Well, in our case, l and w are going to be represented by a different variable, actually. They're both written already in terms of x. I'm going to call this the length and this width. So in our case the length is actually just x plus x, or 2x plus 3. Remember that the m here stands for meters, not a variable. And the width is just 2x. If we want to apply this out we get 4x squared plus 6x squared meters. So instead of writing the area as a function of length and width, instead we're writing it as a function of x now. It's function that only involves only one variable since we had both of these original variables in terms of the same third variable.
Now that Athena has a way of expressing area as a function of x or this half link of the width of the room. She has to start thinking about different values of x to actually come up with a number that this area is equal to. So as a review from last lesson, let's practice plugging in a few different values of x and see what the area of the Fun Room would be. Please fill in the proper values for the output of the function in these green boxes.
A of 6 is 180 meters squared. A of 1 half is 4 meters squared. A of 0 is 0 meters squared. And A of negative 3 is 18 meters squared. Just to be clear, you only needed to fill in the numbers, not the meters squared units.
Now that we have four possible areas for the Fun Room, I'd like you to consider which of these actually make sense in the context of the situation we're talking about. You can always reference the parent function up here, a of x equals 4x squared plus 6x, to remind yourself of what the independent variable stands for in this situation and with the dependent variable is. So, consider both the input and the output in each of these four cases. And please check off which choices that you think do not make sense in the context of Athena's story.
It doesn't seem to make much sense to have a room that is 0 square meters in area. So I'm going to say that A of 0 equals 0 does not make sense in the context of this story. We know that Grant definitely wants a room to exist. His employees want to have a lot of fun. We need actual space to do that in, 0 just doesn't really make sense. We also have A of negative 3 equals 18 meters squared. In this first one that we marked off, neither the input nor the output really makes sense. But in the second choice, it's a little bit trickier because the output seems totally fine. Grant could conceivably have a room that was 18 meters squared in area. However, since x is a length, it doesn't really make sense to plug in negative 3 for it's value. What kind of ruler could you use to measure a negative length? So this trace doesn't really make sense either.
The last quiz then, brought up the subject of what values x can take on in this situation in order for it to make sense. So think about what constraints an architect would logically place on this variable x in this situation. And then write out the possible values that we could plug in for x here using an inequality notation in this box.
All I really know is that x needs to be greater than 0. Since as we discussed earlier, negative lengths don't really make sense when we're measuring walls. And having a side length of 0 also doesn't make sense. X really does need to be a positive value in order for Athena's project to go anywhere.
We just decided that for this situation, we need to have x being greater than 0. But what if I told you that this restriction on x had a special name? In fact, it does. In this case, x is greater than 0 is called the domain of the function A of x. So, if this is the case, then what do you think we can say in general about domain? Is the domain of a function the set of all possible inputs? The set of all possible outputs? The set of all possible values at the independant variable? Or the set of possible values of the dependent variable?
As we said before, the reason we need to restrict what x is allowed to equal is because certain values of x aren't going to make sense in our situation. So the domain here is the set of values that are possible inputs of a of x. So the first choice is correct. We can also think of this, however, as the set of possible values of the independent variable. Since the independent variable and input are the same thing.
When Athena talks about the possible area's of the fun room to other people at the architecture firm, she's probably not going to want to talk about any values of x that aren't allowed. She'll really only want to focus on the ones that are in the domain. In the set of allowed possible values for x or inputs. As we said earlier, the domain of this function is x is greater than 0, because of the restrictions placed upon it by the real life situation it's representing. As we've learned throughout the class, one of the best ways to visualize most things in math is through graphing, and domain is no exception. Looking at what I have plotted here, we can see that the only points on our graph have x coordinates that lie within this domain. It's a little bit hard to see here, but rather than having a filled in circle at the point 0,0 there's actually a tiny little open circle, just like we saw in number lines before. The fact that there's an open circle at this end point is just showing that this end point is not included in the domain. If instead we've had x is greater than or equal to write our domain in interval notation. X lies in the range 0 to infinity not including either end point.
First thing's first, we noticed that both endpoints that we have are filled in. So I'm going to need to use square brackets for my intervals to show the inclusion of these points. Now, the point furthest to the left, in other words the smallest x value that we have on this graph, is negative 4. So that's the lowest value of the domain for which the function has a value. Now, moving as far right as we can along our curve. We come to this point, whose x coordinate is 6. So our interval goes from negative 4 to 6, including both endpoints. We can check to make sure that our interval is right by testing a point outside of the interval and seeing if the function has a value at that x coordinate. So let's say, I pick negative 8. That's right here and there is no function drawn right here. I can't tell you what the function would equal. If you asked me what the value of the function was at negative 8, I could not give you any sort of reasonable answer. The function is just not defined for the input. In fact it's not defined for any input other than the x values of all the points here.
We can also have functions where the n points are not included. For example, in this graph you can see that there's circles at either end point. Open circles other than closed circles, showing that neither of the end points are included in our domain. So in this case how would you write the domain using interval notation?
Our domain goes from 4 to 7, not including either 4 or 7. In other words, x can take on any value between 4 and 7. Remember, when we're looking for a domain on our graph, the only thing we're considering is x coordinates, for which there is a point on the function. The y values here don't matter at all.
So for a given function, we've learned that the domain of that function, is the set of all allowable inputs, the set of all of the values of the independent variable that we can plug into our function to come up with a value. But if we were to talk about the allowable input so much, it's also probably important for us to learn about the possible outputs, the values that the function can equal. The set of all possible outputs of a function is called its range. And how might you find the range? Well, we can figure out what the range is by plugging in all the different values of the independent variable that are in the domain. Once we've covered all of those, we will have covered all of the values in the range. So, the domain then is associated with the independent variable, like x in this case and in many other cases. And the range is associated with dependent variables like f of x. For each of these different variables, the domain and the range basically state what values the corresponding variables can take on. So, considering this definition of the range, what do you think the range of this function is? Remember, this is just the set of all the different values for the function, all the different vertical coordinates expand in this graph itself.
On our graph, the y values range from on the lower end negative 1 to on the upper hand positive 6. However, we know that neither negative 1 nor 6 is included in our range, because the points that these values come from have those open circles on them.
I'd like us to look at one more entirely new function. And for this green curve, please tell me what the domain is and what the range is. Put both of your answers in interval notation. And be sure then that you keep track of whether or not either end is closed or open.
Let's start off with the domain. Our lowest x coordinate of any point on the green curve is over here at negative 8. And we can see that this dot is filled in, so we need an inclusive bracket there. The highest in x that we'd move is over here but we have an open circle at x equals 9. Since we write a 9, but we write an open bracket to show that 9 is not included. So, now for the range. Let's look at the graph from this direction, then. The smallest value for f of x for any of the given inputs that are possible, in this function is -3, and that point is not included because of the open circle. The highest value for y is at y equals 9. So our range, then, goes from -3 to 9, not including negative 3, but including 9.
Now, that we're more familiar with the concept of domain, let's look a little bit more closely at this graph. Our initial function for the area of Grant's fun room as a function of half of its width was A of x equals 4x squared plus 6x. And we said that our domain is x is greater than 0. Now, what do you think then the range values will be? We'll write this in inequality notation as well. And I've already written in A of x for you, so all you need to fill in, in this box is some sort of inequality sign and a number.
Interestingly enough, in this case the range is also just values that are greater than 0. Since the endpoint that we are starting from is this open circle at the origin. And that all of the points lie to the right and above this point. No value of a of x is ever going to drop below the x axis. We also have to know about the overall shape of the entire function, a of x, without the domain restriction. So if you're looking at this in an idealist environment. Maybe it would help us out to look at this entire function without the domain restriction or the range restriction for just a second.
So here's the same function as the function that we had for calculating the area of Grant's fun room, except I've removed the domain restriction, and therefore also removed the range restriction. So the first question I have for you about this entire graph is, what is the domain of f of x? Please write your answer interval notation.
The domain of f of x equals 4x squared plus 6x is negative infinity to infinity. In other words, we can plug in any real number, literally any real number, in place of x, in this function, and we'll get a valid value for f of x. Another way we could express this interval is to say that the domain of f of x is all the real numbers. Remember that whenever we write an infinity sign, whether it negative or positive, we need to have an open parenthesis.
Well, now that we know one thing that the range of f of x is not, it's not all the real numbers, I'd like to know what it actually is. Please, in integral notation, write the range of f of x right here. Remember the coordinates of the vertex that I told you?
So, it is the range then if it is not all the real numbers. Well, we know that the minimum point the lowest value that's a member of the range is negative 2.25 and because the graph actually passes through that point and includes it you need a square bracket. Following the graph in either direction though,we can tell that the values of f of x are just going to continue to increase as the values of x increase. And the same thing will happen in the other direction as well, as x values decrease, n of x values increase. This increasing is never going to stop, so the other n point of the range is infinity, with that open parenthesis.
We know that the vertex is a special point on a parabola. Because if you remember, there's a line of symmetry that runs right through the center of the vertex, and divides the parabola in 2. We saw that tangent lines on opposite sides of the parabola at the same vertical position have opposite slopes. In other words, if we folded this parabola in half, down this side of the line, the two halves this is divided in two right now, would match one and other perfectly. Let's see if we can make any other general statements about the behavior of the function on this side of the graph versus on this side the graph. For a moment, let's just look at this right half. The corner for the vertex as I showed you in the last problem were negative 0.75 and negative 2.25. So for now we're just going to only consider the domain in the region where x is greater than negative 0.75. As you can tell on the right side of the graph, as x gets bigger, f of x gets bigger as well. So I'd like you to complete this sentence property. On the interval of the domain negative 0.75 to infinity, so in other words, this half of the graph we've been talking about. What is happening to the value of the function? Is f of x increasing, decreasing, or constant? So remember that we read graphs from left to right, so this is asking, what is happening to the behavior of f of x, of the value of the function, the dependent variable, as the independent variable x, increases?
F of x is increasing in this interval. In other words, as x gets bigger as we plug in bigger and bigger numbers, into f of x, f of x itself gets larger, since this expression continues to increase.
Now, let's look at that other half of the graph, the part of the graph that's to the left of this line of symmetry, the axis of the problem. In this interval, which goes from negative infinity to negative 0.75, not including either endpoint. What can you say about the behavior of f of x? Again, we're going to read the graph from left to right. So, I'm asking you, as x values increase, what happens to the value of f of x? Does it increase, does it decrease or does it stay constant?
On this interval of the domain, f of x is actually decreasing. So as you move from one x value to the right, increase in the value of x, the curve here is dropping. The y values or rather the f of x values are decreasing as x values are increasing. So whenever a function is going down or has a negative slope we say that it is decreasing on that interval of the domain.
So, to summarize what we can say about function behavior, or at least this aspect of it. If the value of the function goes down as the value of x goes up, then we say that the function is decreasing. Just like over here where it's sloping down. Now if instead f of x increases as x increases, then we say that the function is increasing. Like in this part of the graph. The third option is that f of x doesn't change at all as x increases. Here, f of x is a constant. So here I've drawn another sort of silly looking graph, and I'm going to divide it into several regions. So for each of the regions of the domain that I've split up right here on the left, I'd like you to tell me if in that range in the domain, the function, which is g of x in this case, is increasing, decreasing, or a constant? Please pick one of these 3 values for each part of the domain.
So, here are the answers. In this first part of the graph, we're moving down as x goes up, so the graph is decreasing. Then f of x starts to increase again as we continue to move to the right, so this must be increasing. Then the value of g of x doesn't change at all as we go here, so it must be constant. Moving up again means we're increasing. And then lastly moving down means decreasing.
So, we talked about domain, we've talked about range, we've talked about how we can impose domain restrictions on functions, depending on the situations in the real world that they're involved in, and we've talked about overall function behavior. However, I still want to dig deeper. Let's take a look at this function, g of x equals the square root of x. Now remember, that we just have a radical sign and no sign in front of it, it's implied that this is the positive square root, this is not a plus or minus thing. Let's take a look at the graph of this function. So, here's our graph of g of x equals the square root of x. Now, I could describe its behavior by saying that it's increasing from 0 to infinity. But let's think back to the domain. So, what do you think the domain of g of x is? All numbers between what and what?
Well, the entire curve sits in this part of the graph, starting at the y-axis and moving to the right. So, that means the smallest x value we have on this curve is 0. And then it's going to continue to increase with corresponding values of the function for every value of x that gets higher and higher and higher. So that means, that domain is going to increase forever in this direction, and the domain of g of x, must me all number between 0 and infinity.
Let's think for a second about why the domain for this function, the square root function, is not all the real numbers. Now you've heard a few lessons ago, about how imaginary numbers allow us to take square roots of negative numbers but, and this is super, super, super important If we only care about real solutions, then taking the square root of some complex number is undefined. We want to be able to graph whatever input and output values we get from our function. And if we have numbers with imaginary components, we can't graph them on this kind of graph. So knowing what you know about square roots, and focusing only on real solutions for g of x. What is the correct domain for the function g of x?
The second one is correct the only difference between the two choices is that 0 is not included in the first interval and is included in the second one. If we plug 0 in for x we get g of 0 equals to square root of 0 which is just 0, so the point zero, zero is included on our graph which we can see right here.
Let's switch things up a little bit. So over here, I still have graphed the equation g of x equals the root of x. But I want us to ignore the graph for a second, or just use it as a general guideline. And instead look at this new function, h of x equals the square root of the quantity x minus 5. Now, what is the domain of h of x this time? Please write your answers as an inequality here instead of an interval notation.
We know that whatever is under the square root side has to be greater than or equal to 0. Because otherwise we'll be taking the square root of a negative number and we don't want non-real solutions in this case. So finding the domain of a square root of x minus 5 is solving an inequality. You have to set x minus [foreign] that's our new domain restriction.
So, so far, we've seen two different types of domain restrictions. Some of them are motivated by common sense, problem solving and real world quantities. So for example as we saw earlier, Athena needs to design Grant's fun room so that the dimensions are greater than zero. The other type is domain restrictions that are motivated by math. So restrictions that we need to impose in order to get only real or properly defined solutions. So an example of this would be avoiding taking square roots of negative numbers.
Before we jump into a bunch more practice problems, let's look at one more instance of a domain that is restricted because of what make sense. Just like Athena couldn't make a room with negative dimensions. Let's look at this function, A of r that describes the area of a circle with radius r. We have the function A of r equals pi times r squared, so if this is our circle that we're considering, the radius is this length, from the center of the circle out to its edge. What I'd like to know is what the domain for this function is. Is it all the real numbers, r is greater than equal to 0, r is less than 0, or r is not equal to pi?
r needs to be greater than or equal to 0. Once again, we're measuring a distance, a length and it can't be negative. So the way we'd write this to remain in interval notation, would be square bracket 0, infinity open bracket.
So we talked a lot about domain and range in this lesson so far, but I want to give you a slightly different perspective on how to visualize them, and how to think about values of the domain through the relation of a function mapping to values in the range. So lets say we have some function f and for three to our values a, b and c, constants not exactly sure what they are, we get outputs of one, four, and five. We know that if we graph this, we would pair the coordinates a and 1, b and 4, and c and 5. And what the function is really doing is taking this input value, playing with it somehow, and spitting out this output. So you could visualize this by writing out values of the domain over here, and then showing that via the function, they get paired with these values in the range. Let's say actually that for this particular function f, a, b and c are the only values in the domain. So if the entire domain of this function is just the set a, b, c, what values are in the range? Please separate any values that are in the range by commas.
The domain, in this case a, b, and c, contains all the values that we can input into the function and get defined output values. Those defined output values are the range. So, the range only contains elements that we get from taking elements of the domain and mapping them over using the function. So, the next function takes in the value a, it processes it, and spits out 1. So, a gets mapped over to a value of 1 in the range. That's one of our numbers. B similarly gets matched to 4, and c gets matched to 5. Since none of the elements in the domain map to either 2 or 3, those two numbers aren't in our range. I just listed them here, so that we had, a collection of numbers to go off of and have some contrast.
Now, once again I've laid out a function for you, in the same format as before. We have one oval that contains some possible values of x, and another oval that contains some possible values of f of x, and some values of x mapping to other values of f of x. So, using this diagram, I'd like you to tell me what the domain of this function is, and what the range is. And as before, if you have multiple answers in either box, please separate them by commas.
The domain is only going to contain values of x for which the function is defined. Let's start off with the domain. The domain is just the set of inputs over here, for which the function is defined. So, the only ones that actually map two values over in the f of x oval are 1, 2, and 4. So, those values together are the set that is our domain. Since 3 and 5 aren't actually paired with anything, over here in f of x. In other words, if you plug these inputs into the function, you won't get a output, they are not part of our domain. Now, our range it's a set of values that the domain values map to. So that's going to be 20 from the 1, the 2 maps to 80, and then 4 maps to 80 as well. But since 80 is already in the set, we don't repeat it. Now remember, this does still count as a function even though there's a shared value in the range between two values in the domain.
So we just spent a lot of time talking about domain, range, and all sorts of functions. I'd like to do a sort of review quiz over what a function really is conceptually. We've talked about this in a lot of different contexts. Solving this would be a good way to bridge the many different things that you're learned about functions. So which of these choices correctly describe functions? You can check off as many as you like. Does a function map each element of the domain to at least one element of the range? Does it map each element of the domain to exactly one element of the range? Does it relate a set of inputs to a set of outputs? Does it have to have a defined value for any real number that we want to use as input? And is a function the same as any equation with two variables?
So, actually only two of these are true. A function maps each element of its domain to exactly one element of its range. And a function also can be seen as a relation between a set of inputs and a set of outputs. However, none of the other three choices are true. Let's look at the first one. If we had checked this one off, we will be arguing that a function maps each element of the domain to at least one element of the range. Now, think back to that soda machine example. The soda machine operated like the function, because for any amount of money you put in, there's only one option for the amount of soda that you could get out. Functions work like beautiful machines in the same way. They're deterministic. A single input can only lead to one single output. So, the second one must be right instead of the first one. Now, we know that this last choice isn't true. Because there are some equations with two variables that aren't functions and we haven't delved super deep into this. But you already know that there has to be a special relationship between those two variables, namely, this one. That the independent variable can only map to one value of the dependent variable, and not all equations with two variables do that. For example, we might have x equals y squared, which is a horizontal parabola. Or we might have x squared plus y squared equals 4, which gives us a circle. Neither of these passed the vertical line test, which we talked about before, so they're not functions. And they don't pass the vertical line test because they don't meet this criteria. And finally, if we said that functions must have a defined value for any real numbers input, then we would be saying that the domain of every single function has to be all the real numbers and that's just not the case. We talked about two different major types of domain restrictions that prevent certain real numbers from being within the domains of certain functions. We'll talk about even more of these cases later on. You just have to wait a couple of weeks for some really, really exciting stuff. I know the past couple of lessons have required a pretty major paradigm shift in terms of the way that we think about the relationship of variables to one another, and what equations really are. But I hope that you've found our work with functions interesting, and I think that it will prove more and more helpful as we continue to use the function paradigm in our problem solving especially in our setting up of equations. Great job.