Pretty much everybody has to pay taxes and Grant, with his new Gleaming Glasses company is no exception. The way that taxes depend on income can be described by functions. But these are slightly different functions than ones we've seen before. Let's check them out. First things first, here's a chart showing how a person's yearly income, determines the amount of money they will pay the government in taxes where Grant lives. We can see that the tax rate over here, or the percent of their income that a person will pay the government in taxes, changes depending on how high the person's income is. The way this works is a little bit complicated, so bear with me here. Now someone makes under $5000, please note this entire column is in dollars, so if someone makes under $5000, their tax rate is 0%. So, they actually don't have to pay the government anything at all. If someone's total income is between $5,000 and $15,000, the first $5,000 of their income is set aside and not taxed. Then what is left over is taxed at this 10% rate. If someone earns between $15,000 and $35,000 total, then, as with the person in this lower bracket, the first $5,000 of their income isn't taxed. The next 10,000, is taxed at 10%, and then the rest is taxed at 20%. If the person makes even more money than this total, then the amount above $35,000, is taxed at 30%. We can think of this tax rate scaling as a sort of layering effect. As were counting up the money a person makes, we don't start by thinking about the total. We start by counting from zero. Once we get to 5,000, then we start to add our tax rate, and we keep counting up and up, increasing the rate when we need to.
So if for example, Grant earns $27,000 total, what is the total amount he's going to have to pay in taxes? First figure out how much of his income falls within each of these tax brackets. Then, I have listed the tax rate for each of those brackets over in this left-hand column. So to figure out how much he's going to pay in taxes from each of those brackets. Just multiply the income that falls within that bracket by the rate of that bracket. Once you have all of the taxes from each of the brackets, add all of them up to come up with the total amount of tax Grant's going to have to pay. I'll write in a dollar sign for you right here, and you don't have to write them in any of these slots at all. Just write the numbers.
So, here's what our chart should look like filled in. As always, Grant is going to pay nothing in taxes for the first $5000 he made since the rate is 0%. Since $10000 of his income falls in this 10% range, he'll have to pay $1000 for that. And the remaining $12000 all belongs in the middle bracket which is taxed at
I think it will be helpful for us to visualize the relationship between, total income and total pay in taxes. So, let's graph it. If we want to graph income on the horizontal axis. In taxes owed on the vertical axis, both measured in dollars. What will the slope of the line we draw represent. Will it represent the tax rate, written as a percent? The tax rate written as a decimal, the total tax paid, or the total income earned.
Since the slope in this case will be the change in tax paid, over the change in income earned, that will give us the tax rate written as a decimal. If we wanted to see the tax rate written as a percent instead, we need to multiply, the decimal by 100.
Now that we have an idea of what our graph will mean, I've gone ahead and drawn it for you. Know that the graph looks like it's made up of pieces of four different lines with different slopes. And in fact, it's exactly what's happening. So, we can see how they all compare to one another. I'd like you tell me what the equation in for each part of the line is. Remember that the slope of each line will be the tax rate of the bracket that corresponds to it written in decimal form. If you figured this out for each line segment, and then simply find a point on each of these sections of the line, you can use the point-slope formula to find the equation of the line. Please use x for the income, which is on the horizontal axis and t for taxes owed, since it's on the vertical axis.
Here are the equations for each of the pieces of our line over here. The first one was pretty easy to figure out, it's just y equals 0. Since this is just a horizontal line at well, y equals 0. For each of these other three equations, I started out using point-slope form and then transformed to have equations in slope-intercept form. Over here on the graph, I wrote the tax rates corresponding to each of the brackets to make reminding myself of the slope of each line really easy. Then I just picked a point for each line that was easy to identify. Plugging in all of that information gave us these equations. For the lowest bracket, we have y equals 0.1x minus 500. For the middle one we have y equals 0.2x minus 2000. And for the highest bracket, or at least the highest one shown here, we have y equals 0.3x minus 5500.
When we put all these pieces together, what we get is actually a function. We can tell, since it passes the vertical line test. This kind of function is called a Piecewise function because, well, it's defined in different pieces. Since different functions are defined in different parts of the domain, there have to be at least two different equations defined in different parts of the domain for a function to be considered a Piecewise function. What I'd like to know, is what part of the domain each of these functions occupies? I'm going to switch over to Piecewise function notation right now so that you can get an idea of what that looks like. So this is what typical Piecewise function notation looks like. You can see that instead of separate equations for each part of the line over here, we've effectively merged them into one using this brace, this curly thing right here. This shows that all of these different function pieces are part of a single function f of x. Now the other important feature of Piecewise function notation, is this section over here to the right. This is where we show the domain that each part of the function occupies. Often, you'll see a comma used between the equations for the different parts of the functions and their domain. But sometimes you'll see the word for written in here instead like this. To save space, I'm going to use commas right now. But either way is acceptable. Now, looking at the graph, and thinking back to what you learned about Grant's tax brackets before. Please fill in the different numbers that belong in these boxes to show what the domain of each piece of the function is. Now I've been nice, and I've already filled in the inequality signs for you, so all we need here are the numbers.
Here are the sections of the domain that correspond to our different parts of our piecewise function. Note that each of these domains directly corresponds to the income ranges that went along with the different tax brackets. Just like we saw on the chart earlier on. For the first piece of the function, we have zero is less than or equal to X is less than 5000. Then we switch to this new equation whose domain is 5,000 is less than or equal to X is less than 15,000. The part of the domain for the next piece of the function is 15,000 is less than or equal to X is less than 35,000. And finally the highest tax bracket we have listed here is for all values of X greater than or equal to 35,000.
Now are linear equations, the only kind of functions that can make up the pieces of a piecewise function? Definitely not. We can go crazy and make each section any kind of function we want. So, how about this graph? How would we express the piecewise function that I've shown right here? Please fill in both the function part in the domain part for each portion of the graph. Now when you do this, I'd like you to start at the top here with the left-most portion of the graph, and then work your way right. That way the values we see here in the domain are going to increase.
The function we see over here on the left, is a line that is a slope of negative one half. And its y intercept is negative 4. So, it's equation is negative 1 half x minus 4. We can see that it starts here at x equals negative of the graph isn't a straight line however, it's a Parabola. That's pretty neat. And note that in order to use the notation we've been using for piecewise functions, we have to rearrange the equation of each part of the function so that f of x is by itself on the left hand side. If we do that for this parabola, it's actually not too bad. It's just negative the quantity x minus 2 squared, since the vertex is at 2 comma 0. The graph is obeying this function from 0 to 3. The last line we have has a slip of 1, and it's equation is just f of x equals x minus 4. It's on the graph from 3 to 8.
In all of the piecewise functions we've looked at so far, the endpoints of each of the segments have matched up with one another exactly. Which makes the graphs continuous. There are no breaks. Now when this is the case, we can choose to include these boundary values. These values of x where we switch from one part of the function to the other. To be in either of the two parts of the function that come together at that point. We don't want any over lapse however. So in the case of this last function we were looking at. We initially included that x equals 0 point. In the probable part of the function. Which is why we had 0 is less than or equal to x is less than 3. As the domain, for this middle piece of the function. We could however instead, include 0 in this first linear part, the negative one half of x minus 4 part of the function. If we decide to pursue that, we'd have to switch off our inequalities over here. So instead of having 0 is less than or equal to x right here you have 0 is less than X. I would switch the equal to part to be here over. So domain for this first line is negative five is less than or equal to x, less than or equal to contained in the f of x equals x minus four portions. This line right here. We could instead, include it in the parabola if we just made x is less than or equal to 3. Instead of x is less than 3, and if we just switch this to less than sign. Sometimes however, piecewise functions aren't continuous. They often have breaks in them. But sometimes Piecewise Functions aren't continuous. They often have breaks in them, as you can see in the two functions I've drawn here. Now, if you want to show that a certain point is not defined on a function, or in other words, if you want to exclude its X coordinate from the domain. We just draw a little open circle at that point like I've done in several places here. Now, for each of these two piecewise functions here, I'd like you to decide which of these end points should stay open. And which ones should be filled on, based of course on how I've defined the domains for either of the functions. If you think an endpoint should be filled in, just click the circle that's at that endpoint. And if you want it to stay open, then just leave it open. Don't mark it. To double-check your work, you can make sure that there's no x value on either of these graphs that belongs to two different parts of the functions.
For this graph, the point x equals 1 belongs with the 2x minus 1 part of the function, since its domain is x is greater than or equal to 1. Over on the right, we should have this end point and this end point filled in. Since we have f of x equals negative 2, for x is less than or equal to negative 3. And, we have f of x equal 5 minus x, for 4 is less than or equal to x. Note that the middle piece of the function, 1 half x minus 3, has an inequality that isn't inclusive for either of the numbers involved. So, that means, it should not include negative 3 or 4, which we see with these open circles here. Awesome.
Even though a piecewise functions probably feel unfamiliar at this point. You've actually met a function before which can be written as a piecewise function. It's the absolute value function, remember that it has this V shape Now how can we define this as a piecewise function? Fill in the function that's obeyed when x is less than 0 and the one that's obeyed when x is greater than or equal to 0. Just a hint, you cannot get away with putting the absolute value of x in both of these boxes. I want to know what different functions show up in these two different regions.
Instead of writing f of x equals the absolute value of x, we could write f of x equals negative x for x is less than 0, and x, for x is greater than or equal to 0. Notice that writing the function in this piecewise form actually tells us what it means to take the absolute value of a number. The absolute value signs switch the sign of negative numbers and preserve a value of positive numbers. That's pretty cool.
I don't know about you. But I think that online shopping is pretty great. Once Grant realized how much easier many people find it to be, he decided that he should make his glasses wipers, nozzles and cleaning solutions available for purchase on the internet. To encourage his customers to buy more merchandise online, he's come up with a fancy piece for his function that allows them to save money on shipping cost orr postage, if they spend more money buying his products. Please note that costs of order here does not include the postage. And also note that the quantities in each of these columns are measured in dollars. So, we can read this as, if the cost of the order is less than $10, the postage is only $2.75. If the order costs between $10 and $50, postage is $5.00. And for a person who spends at least $50 on the site, they need not have to pay any postage at all. How much would the total cost, which is equal to the cost of the order plus the postage, be for each of the following. A, one nozzle set for $9.99. B, a box of six nozzle sets for $49.99. C, some cleaning solution for $1.99 and one nozzle set for $9.99. And d, a box of six nozzle sets for $49.99 and some cleaning solution for $1.99. I've included the dollar signs for you on all these amounts, so please just write in the numbers.
All we need to do for each of these, is to figure out what the cost of the order is. Then what the corresponding cost of postage will be. And add those two quantities together to get the total cost. If a nozzle set costs $9.99, and that's all were buying, that's under $10. So postage is just $2.75. $9.99 plus $2.75 equals $12.74. For B, if we do the same thing, our postage is going to be $5 instead, and we end up with $54.99. For C and D, we need to add together two quantities to find the cost of the order, and our total cost for these two end up being $17.98 and $51.98.
Here I've drawn a graph showing how postage, which I'm going to call f of x, depends on the cost of the order, or x. Note that like a couple of other piecewise functions that we work with so far, this function has breaks in it. The different parts don't connect to one another. Because of this, we call functions like this one, discontinuous. What I'd like to know is what piecewise function this graph is representing? Please fill in the different parts of the function for each of these parts of the domain right here. Of course, you can always use our table for reference, too.
The three functions we see graphed here, to make up our single piecewise function are f of x equals 2.75, for 0 is less than or equal to x is less than x equals 0 for 50 is less than or equal to x. These correspond directly to the different postages depending on how much our orders cost.
This last function that we looked at is a bit of an interesting kind of piecewise function, it's called a step function. A step function, is a function that's made up of one or more constant functions, as you can see for the last one we worked with. In both the graph and the equation, each piece is in fact a constant function. f of x is equal to just a number in each part of the domain. And we have a bunch of straight horizontal lines. In light of what you've just learned about step functions, which of these graphs do you think represent step functions? Remember that a step function is just a function made up of one or more constant pieces.
These two graphs are the only two that show step functions. The other two aren't, because this one has linear parts, with slopes that aren't just 0, and this one has a part that's a problem. This third one is sort of a funny case, you might call it a trivial case of a step function, because it's just one constant function. This is just f of x equals 3. This step function only has one constant piece, so it's sort of a silly example, but it still counts. This first one is a more classic example of a step function. And probably something you'll see more typically when people talk about step functions.
Sales of Grant's packs of 12 nozzles have really taken off. In fact, he's been doing so well selling these bulk packages, that he's only going to sell nozzles by the dozen now. That does mean, though, that depending on how many nozzles he has total, he might have some left over. For example, if Grant has 40 nozzles total, then that will make 3 and 1 3rd packages of 12 nozzles. Since if we take packages of 12 nozzles, but then there's this 1 3rd of a package left over. Grant can't sell 1 3rd of a package. So, this 3 and a 3rd packages really only counts as 3 for him. So, it's almost as if we're taking this 3 and a 3rd and rounding it down to the nearest integer, 3. So, let's say that x equals the actual number of packs of nozzles we have. So, for example in this case, we have 3 and 3rd packs of nozzles. And let's say that y equals the number of packs of nozzles that Grant can actually sell. So, in this case, that was 3. We can represent this relationship between the actual number of packages we have and the number Grant can sell by something that's called the Floor Function, also known as the Greatest Integer Function. The notation for the equation is this, y equals x inside these double square brackets, which basically means that the y value is equal to the greatest integer that is less than or equal to the corresponding x value. But what do you think the graph of the 4 function looks like? Which of these graphs do you think properly represents the floor function? Remember that we want the y value to round x down to the nearest integer. I know that from our way it might be hard to tell these graphs apart. So, below them I've shown a blown up version for you of what each of these steps looks like. Please notice the difference between the 2 in terms of which circles are empty and which circles are colored in.
And this one is the floor function. You can see that for a given x value, the y value will be equal to the next lowest integer. Unless x itself is an integer, in which case y equals x. Please know that the only difference between this graph and this one, is the placement of the empty circle and the closed circle in each step.
By now, we've seen a ton of different kinds of functions in this course. We've seen polynomial functions, absolute value functions, square root functions and our piecewise functions, too. One thing we haven't looked at though is functions that involve division of polynomials, functions that have fractions in them. What kind of equation would a function like that have? Well, here are a few examples. All of the functions I've written here involve taking ratios of polynomials. In other words, dividing one polynomial by another. Remember, that even constant terms are considered polynomials, too. We call them rational functions because they are ratios. There's a ton of variety in this family of functions, and their graphs might not look quite like you expect. So, here are the graphs of the four functions that I listed up here. But now they are in a different order. Figure out which function each of these graphs represents and then write the letter of the graph next to the proper function. I know that we've never actually seen rational functions before, so to guess what each of these look like, you can plug in some different points and then see which of the graphs they fit.
Graphs A and D probably look pretty familiar to you and they go respectively with functions G and F. I know it probably seems a little bit strange because we haven't seen equations like these before, even though you recognize their graphs. We'll talk more about this later, but start to think, right now, about what these equations reduce to. And why they produce the graphs that they do. Graphs B and C however are definitely new for us. Graph B represents the function one over X and graph C represents the function one over X squared. You'll see a lot more graphs that look like these two over the next few lessons. And pretty soon you'll understand exactly what's happening in each of them.
So let's pick one of these graphs to play with a little bit more so we can get better acquainted with rational functions. As I'm looking at these equations, I think that h of x is equal to 1 over x would be a great one to start out with. So, let's pick that, I'm going to blow it up here for you. So here's our graph blown up. Now there actually are two lines for the graph of y equals 1 over x will never cross. Can you tell from the graph here, or from the equation, what those two lines are? Please type in their equations right there. Now, don't worry at all if you don't get them. I'm going to talk a lot more about this, in just a bit.
The two lines of this graphs is never going to cross are y equals 0. The x axis over here and x equals 0. Now I didn't draw these two lines perfectly. But in reality the graph of y equals 1 over x just gets super, super close to both of the axes. But it never, ever touches them. That's really cool.
Pretty soon we're going to look at some even more interesting functions, ones that involve absolute value signs. But I think that before we do that, we should do a little refresher on absolute value itself, or absolute value functions. I've written down several equations for you and also drawn several graphs. Please match the letter of each graph with the proper equation.
Graph C and B are pretty easy to identify, since they're just linear equations. So, they go with h of x and k of x. A and D are the two absolute value functions, so they are the functions f and g. We can figure out which one is which by plugging in a few numbers. Let's say, x equals 0, for example. For f of x that gives us f of 0 equals the absolute value of negative 3 or a 3, which is what we see in graph D. The same thing with g gives us g of 0 equals negative 3, which is going with graph A.
Let's see how to write the two absolute value functions from the last quiz as piecewise functions, this is just like we did earlier on in the lesson. This time however, I'd like you to fill in both the function pieces and the corresponding parts of the domain.
For g of x equals the absolute value of x minus 3. We can write this in piecewise form as g of x equals negative x minus 3, for x is less than 0, this branch over here and g of x equals x minus 3 for x is greater than or equal to equals negative x plus 3 for X is less than 3, and x minus 3 for x is greater than or equal to 3.
Remember these four functions that we played with at the very beginning of our discussion of Rational functions. Well, we're going to play with them a little bit more right now, except I'd like to make one change to each of them. I want to add in some absolute value signs. Take a second right now to predict how the addition of these absolute value signs to each of these functions is going to change the way that their graphs look. Now that you've had a chance to make your prediction, here are the graphs of our functions. You however, need to tell me which graph goes with each function. Pick some points to test out and then put the letter of the graph next to the function it belongs with.
So here are the graphs that match up with each of the functions. You might notice that two of these graphs actually didn't change at all. From the way they looked when their corresponding functions didn't have absolute value signs in them. Those two graphs are B and C. So what does this mean? Well, you'll note that in both functions g and k, the absolute value signs appeared around x squares. Now we know that x squared on it's own is always going to be positive. So taking the absolute value sign of a positive number doesn't do anything to change it's value. That's why these two graphs don't change at all, compared to the old versions. We put absolut value signs around the denominatro of the function 1 over x. We can see that the branch that used to be down here with negative x and negative y values. Got flipped across the x axis. So now, it's y values are positive. I think that graph d might actually be the most interesting change of these four. Before, we just had a straight line. This is basically just the function. F of x equals 1. A constant function. But now, our numerator will always be positive. But, once we move over into the negative x values. The denominator will be negative. A positive divided by a negative makes a negative, which is why this half of the line got reflected across the x axis to become negative. Pretty cool.
Many of the functions we've been dealing with recently then, are rational functions. A rational function is just the ratio of two polynomial functions. Let's say for example, that we have the function f of x equals x squared plus for a second. We can factor this. If we do that, then what should we have multiplied together in the top of the fraction?
x squared plus 3x minus 10, factors to be x plus 5 times x minus 2.
Considering the function that we have, what real value of x can we not substitute in to f of x? In other words, what values can x not equal here?
x cannot equal negative 5. If it did we would end up dividing by 0 since the denominator would be negative 5 plus 5. This means that negative 5 is not part of the domain of f, since the function is not defined here.
Now, as with any fractions, we have a common factor in the numerator and the denominator. We can divide them by that factor and cancel it out. I'd like to find the greatest common factor shared by the top and bottom of our rational function here and get rid of it to write the function in simplest terms. Please note that I've included our domain restriction over here, so you don't have to worry about writing that in.
Since we factored the numerator, we can see that we have a common factor of x plus 5 in both the top and the bottom here. So, if we cancelled that out, the only factor that's left is x minus 2 in the numerator. That means f of x reduces to just x minus 2. Of course keeping in mind that x is not allowed to equal negative 5. Now on our graph when x equals negative 5, we're just going to see a tiny little point size brick, that's all this domain restriction does. It's pretty amazing considering what a complicated function we started out with to see that it reduces to just x minus 2, something linear. Rational functions are very fascinating.
Let's see if a similar sort of reduction happens with another rational function. How about we play with this one? F of x equals x squared minus 1 over denominator of this function? Please just factor for now. No cancelling anything out yet.
The numerator is just a difference of 2 squares, so it's equal to x plus 1 times x minus 1, and the denominator, a factor of 2, has already pulled out for us. So x squared minus 3x plus 2 factors to be x minus 2 times x minus 1.
Now that we have our function in its fully factored form, is there anything that we can cancel out? Please write the reduced version of this function, getting rid of the greatest common factor in the numerator and the denominator.
We notice that there's a common factor of x minus 1 in the numerator and the denominator. If we cancel those out, we're left with x plus 1 in the numerator and 2 times x minus 2 in the denominator.
Let's take a look at the graph of the function we were just dealing with. Here it is, there are two lines that this graph never crosses. Which two do you think they are?
Once again, we have a rational function with a fascinating graph. It never crosses the line x equals 2, or the line y equals 1 half. Let me take a second to draw them on here. Here we have the line x equals 2. We can see that on either side of it, the graph is getting closer and closer. But it's never going to actually touch the line. This is called a vertical asymptote. Let's look at y equals 1 half, and there we go! This is called a horizontal asymptote. Similarly, on either side, the graph is getting closer and closer but never actually touching it. We're going to talk a lot more about these kinds of asymptotes and others in the next couple of lessons.