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Contents

Notes for Lesson 28: Inverses

Cleaning Solution

Many, many lessons ago we helped Grant choose between three different brands of glasses cleaning solution to sell along with his wiper sets. One brand that seemed like a pretty good option gave us this equation, y equals 10x plus 150. In this equation, x stands for the number of gallons of cleaning solution that Grant orders, y is equal to the total cost of the order. The solution cost \$10 per gallon. And there is a flat delivery fee of \$150 added on, regardless of how much solution was ordered. Before, Grant decided how much solution to order, so what x should be based pretty much just on how much he wanted or needed. When we plug that number in for x, the function would output how much it would cost him. However, Greg's being a bit more strict with his budget these days. And instead of doing things the old way, he wants to set aside a certain amount of money to spend on cleaning solution. That means that now, we probably need to switch up our equation a bit. He's going to want to calculate the amount of solution ordered based on the amount of money spent. Since now, amount ordered depends on money spent, let's switch up our variable names. If we want x to equal the total cost of the order and y to equal the amount of cleaning solution ordered, what will our equation look like now? Remember that all of our quantities are still related to each other in the same way, we're just changing their names.

Cleaning Solution

All we need to do is switch the position of the x and the y in the earlier equation. So now instead of y equals 10x plus 150. We have x equals 10y plus

Strange Equation

It's great that we have this new equation with variable names that are more appropriate for our situation. But this equation does look a little bit strange. We're talking about y, the amount of solution ordered. And how it depends on x, the total cost of the order. So typically, we would want y by itself on one side of the equation to show exactly how it depends on x, mathematically. If we make that happen, what will be on this other side of the equation?

Strange Equation

Modifying our equation to get y all by itself on one side of the equation gives us x over 10 minus 15 on the other side. There are of course several different ways you could write this. You could have just said 1 over 10 times x minus

The equation we just came up with, y equals x over 10 minus 15, and the one we started out with, y equals 10x plus 150, are related to one another in a special way. To figure out what that is, I think it would be great if we plotted both of them. To get to that point, though, we need to know, well, some points on each line. Please fill out each of these x y charts to give us some of those points.

Here are our filled-in versions of our tables. Looking at the numbers we wrote in for y here, and comparing them to this table, and then looking at these numbers for y. And comparing them to this table, there's definitely something pretty interesting going on here. Let's take a peek at the graphs of these functions to maybe get a better idea of what that is. Here are the graphs of our two functions. This red one shows y equals 10x plus 150, and the purplish one shows y equals x over 10 minus 15. Start to see if you can pick up a pattern relating the points on this line to the points on this line, using both the tables and the graph here.

Points on the Function

The course that you just did can help us start to see the connection between our two functions here. Let's see if we can really pin down this relationship is though. If we have a point a,b that we know allies in the function where y equals 10x plus 150. Then do we know anything about a point on our other function, y equals x over 10 minus 15. Please pick the statement that accurately completes this sentence.

Points on the Function

This second answer here is correct. If we know we have a point a,b on this function, then the point b,a will lie on this function, we can tell that from the graph right here. We have the point 0,150 and the point 150,0. These points just have their coordinates switched from one another. The same is true of all these other points we have graphed. If we take a point on the red line and simply switch its coordinates, we end up with a point on the purple line. That is really cool. Just looking at these two equations, you might never have expected that kind of close relationship.

Inverse Functions

The relationship between these two functions we've been talking about has a special name. They are inverse functions of one another. As I said before, just looking at these equations, you might not have any idea right off the bat that these functions are inverses. To help make that more clear immediately, there's a special notation we can use. Let's say that we have a function that we're calling f. Then if f has an inverse, we can write that inverse like this, f with a little negative 1 up here to the right. Now we have to be super careful, when we use this notation. This does not, I repeat, does not mean 1 over f. The negative 1 here is not an exponent. You read this as f inverse, not f to the negative 1. To make sure that we're keeping everything straight. Now, let's suppose that f, in this case, is f of x equals 3x minus 1. Right away, we can plug in values for x to figure out different points on the line. I'll just pick out a few for right now. We have 0 comma negative 1. 1 comma 2, and negative 1 comma negative 4. We don't know the equation for the inverse of f quite yet, but we do know that we can denote it like this, and we actually know quite a bit more information about this, because of these points here. Based on the information we have about the function f, what points do we know will lie on f inverse? Below each of these points that lie on f, Please write the corresponding points on f inverse.

Inverse Functions

To figure out each of these points, we just need to look at the point above on f and switch the order of the coordinates. So instead of 0, negative 1, we'll have negative 1,0. Instead of 1,2, we'll have 2,1. And instead of negative 1, negative 4, we'll have negative 4, negative 1.

Finding the Inverse

As great as those points we found are, I'd still like for us to find the equation of f inverse. How about we take this through step-by-step, since this is our first time really doing this. We know that we need to start out with this equation, the equation for f. As we've seen from graphs and points in equations before, we know that to find the inverse, we need to switch x and y in our equation. However, we don't even have a y in our equation right now. Let's get that straight right now. Please replace f of x in this equation with y, and then rewrite the equation down here.

Finding the Inverse

That gives us y equals 3x minus 1.

Switch x and y

Since the inverse of a function will have its x and y coordinates switched from how they were in the original function. The x's and y's must be switched in this equation, too. How about we do that next then? Please switch the position of x and y in our equation y equals 3x minus 1. Write the new equation in this box.

Switch x and y

Switching x and y here leaves us with x equals 3 y minus 1.

Isolate y

This equation looks sort of funny though. Since y is the dependent variable, we tend to write it by itself. With an expression in x on the other side of the equation. Lets make this happen here. If we have y on this side, what belongs in this box to complete the equation? Remember, we're working off of the equation x equals 3y minus 1.

Isolate y

Starting with x equals to 3y minus 1, we can add 1 to both sides. And then, divide both sides by 3 to end up with 1 3rd, the quantity x plus 1 is equal to y. Then, of course, we can just switch what's on either side of the equation. That way, y is on the right side of the equation, and we have y equal 1 3rd times x plus 1.

Rename y

We've come a long way. Now we have the equation y equals 1 3rd times x plus 1. How does this equation relate to the one we stared out with? Well, it's it's inverse. Let's make that a bit more obvious though with our notation. Please replace y in this equation with something to show that this is the inverse of f.

Rename y

If instead of y, you write f inverse of x, then we're showing that this function is the inverse of this function.

Find Three Inverses

Let's take a second to step back and look at the steps we've gone through over the past few quizzes to move from f of x, to f inverse of x. First, we change variable names from f of x to y, just to make our manipulations a little bit easier. Then we did something that's generally not legal when we're modifying equations. We switch the positions of x and y in our equation. This is a really important step, because it's in this step, that we're actually switching from the original function to the inverse function. After that, we rearrange our variables once more, so that y was by itself on one side. Then since we know this is the inverse function. To make that clear, we change the name of y to f inverse of x. Showing its relationship with the original equation we started out with. I think it would be worth it to have a bit more practice finding the inverses of functions. So let's do that, I've written three functions here for you, f of x equals x plus 1, g of x equals one fourth times x minus 9 and h of x equals x cubed. For each of these functions, I'd like you to find its inverse. If you're having trouble remembering what to do and why, then please go back and just watch the past few videos, or just review the beginning of this quiz.

Find Three Inverses

The three equations we have here in black boxes are the inverses of the equations we started out with. I've written out the steps to find each of these inverse functions, and there's one important thing I'd like you to notice. To indicate the point at which we switch from modifying the original function to modifying the inverse function, I changed colors. So the first two steps of each equation are still working with the original functions. Then at the point where we switch the position of x and y, we've changed the equation to be the inverse function. All the work with the inverse function, for each of these, is in purple. The actual steps that we've gone through when modifying either the original function or the inverse function are pretty standard algebra. They're things that you learned about very early on in this course. Why don't we take a look at this middle one just as an example. As we discussed earlier, the first thing that I did was switch from calling this g of x to calling it y. Just so that when we're moving things around, it's easy to tell what that variable actually is. We won't get it confused with anything. Then, to change from the original function to the inverse, I just switch this box of x and y. Since that if y equals 1 quarter times x minus 9, we have x equals 1 quarter times y minus to get y by itself. Since I want g inverse of x on the left side of the equation, since that's our standard function notation, I just took one extra step to switch the sides of the equation. So that y was on the left instead of on the right. That gave us a final answer of g inverse of x equals 4x plus 9. The other two answers we ended up with were f inverse of x equals x minus 1. And h inverse of x equals the cube root of x. Now instead of writing cube root x, you could of course have written x to the 1 3rd power instead. Either way is correct. I hope this extra practice with inverse functions helped you feel a bit more comfortable with the process we have to go through for finding them.

Find Three More Inverses

I think we're really on a roll with this whole finding inverses thing. So lets just keep it up. In fact, I'd like us to start with the equations that you just came up with. The ones that were the inverses of the ones we started out in the last quiz. I'm going to give them new names though. So now I'm going to call them the functions p, q and r. We have p of x equals x minus 1. q of x equals these functions?

Find Three More Inverses

So here are our three answers. We have p inverse of x equals x plus 1, q inverse of x equals one fourth times x minus 9, and r inverse of x equals x cubed. I've done the work in the same way as last time including the color change, when we switch from the original function to the inverse function. And what's really important to notice about this quiz is that the equations we end up with in the end. As the inverses of these functions, are the equations we stared out with in the quiz before this one. We started with f of x equals x plus 1, and found that it's inverse was f inverse of x equals x minus 1. We ended up with this, 4x plus 9, by taking the inverse of a function that was equal to this one. And the same is true of this equation. I'd like you to mull this over and keep it in mind.

Inverse Features

If we have a function f, and we know its inverse, f inverse, then what else is true? Is it true that f and f inverse are inverses of one another? Is it true that f is the inverse of f inverse, or do we not know anything else for sure? Please pick as many choices as you think are correct.

Inverse Features

As we saw in the quiz before this one, if we are 100% positive that f inverse is the inverse of f. Then that also means that f is the inverse of f inverse. This may just seem like symantics but that also means that f and f inverse are inverses of one another. In other words, if we start with a function f, and we take it's inverse we end up with f inverse. Then if we take the inverse of f inverse we just end up with f. This explains why in the last two quizzes we did, we circle back to end up with the equations we started out with.

Reflected in what

We've been talking about equations for functions that are inverses of one another, and also points that lie on them. And we have seen one graph of two inverse functions, but I think that it would be great if we could focus on the graphical part a little bit more. So, here I've drawn for you two functions that we were working with earlier. We started off with f of x equals 3x minus inverse. But, I'll keep with this notation for now. This is pretty tricky, but how are these lines related to one another? I'll give you some choices to make things a bit simpler. Which of these choices properly completes this sentence? The graph of f inverse is the reflection in the what of the graph of f? Your choices are x-axis, y-axis, or the line y equals x. Think about what it would look like to reflect f in each of these lines, and then see if one of those pictures matches the graph of f inverse.

Reflected in what

The graph of f inverse is the reflection in the line y equals x of the graph of f. If we draw the line y equals x on our graph over here, we can see that it cuts the picture in half. This half is identical to this one, just a mirror image. This part of the original function also matches perfectly with this part of its inverse. The statement that we have over here that f inverse is the reflection in the line y equals x of f is actually true of any function of inverse, not just these two lines. This is a result of the fact that inverses are the same equations just with x and y switched. It always amazes me to see how relationships between graphs reflect relationships between equations.

Composition of Inverses

Since you had so much practice with composite functions, in that last lesson. I wonder what would happen if we made a composite function by using a function and its inverse? These functions have been working pretty well for us so far, so let's just keep going with them. Using these two function then, what is f of f inverse of x? And what is f inverse of f of x?

Composition of Inverses

For these two functions, f of f inverse of x, just gives us x. And f inverse of f of x also gives us x. Remember that when we create composite functions, what we're doing is taking the output of the function that's on the inside. And using it as the input for the function that's on the outside. This seems pretty uncanny that we end up with x for both of these compositions. Maybe we're on to something here.

So Many Compositions

Now the question is, is this property something that happens for every pair of functions and their inverses? Do we always end up with the independent variable when we take their compositions? I think we should test a few to see if we can disprove our theory. How about we just use those functions and their inverses that you found before? We're going back to f of x equals x plus 1. G of x equals 1 4th times x minus 9, and h of x equals x cubed, and of course, the inverses that go along with them. I'd like you to find a bunch of different composite functions. I've listed them all down here for you. Please tell me what you get for each one.

So Many Compositions

Lets try one of these out together to see what happens. How about one of these more difficult ones. Let's say g of g inverse of x. g of x is 1 4th times the quantity x minus 9. So in place of x, we need to plug in the function g inverse, which is 4 x plus 9. So, look at 4 4th times 4x plus 9 minus 9. What does this simplify to? Inside the parentheses, we just end up with 4x and 1 4th times 4x is equal to x. Hmm, I'm going to go through and find all these other compositions right now. Let's see what happens. And it looks like each of these composite functions is in fact just equal to x. This confirms another fascinating property of inverse functions. The composition of a function with its inverse and vice versa will both give us what is known as the identity function. It's just the independent variable, which in this case is x. So, another way to think about inverse functions is that the function f inverse will undo whatever the function f has done to the independent variable. In this case, f is adding one and f inverse is subtracting one. Here, g is subtracting Lastly, h cubes and h inverse takes the cube root. I think we're really getting down to what it fundamentally means for functions to be inverses for one another. That's really awesome.

Are they inverses

Suppose then that we have two functions. How about these two, f of x equals x over 4 and g of x equals 4 over x. Are they inverses of one another? Please pick yes or no. Think about how you can use what we just learned to figure this out.

Are they inverses

And the answer is no. These functions are actually not inverses of one another. It might look like they are right off the bat since we've been talking about how inverse functions undo what the other one does. However, these operations performed by f and g are not actually opposites. F divides x by 4, but g of x divides 4 by x. We can tell by taking their composites that they are definitely not inverses of one another. They're definitely not undoing each other's operations. If we take f of g of x, we end up with 1 over x. And if we take g of f of x, we get 16 over x. Hm, I wonder what the inverses of these functions are, then?

Find the Real Inverses

Since these functions aren't inverses of each other, let's find their actual inverses. What is the inverse of f? And what is the inverse of g? Once you think you've found them, make sure that you check your work by finding f of f inverse of x, and all these other compositions down here. Remember what they all need to equal in order to verify that these actually are the inverse functions we're looking for.

Find the Real Inverses

Going through the steps that we know and love now for finding inverse functions, here's what we end up with. Starting with f of x equals x over 4, f inverse of x equals 4x. This makes perfect sense. F is dividing x by 4, and f inverse is multiplying it by 4. Checking our work, f of f inverse of x equals gives us 4 times x over 4, which is just x. Awesome! This must really be our inverse. Looking at g of x, we started off with g of x equals 4 over x. And interestingly enough, g inverse of x also equals 4 over x. Let's double-check to make sure this works. G of g inverse of x is 4 over 4 over x, which simplifies to x. Hm, we get exactly the same thing for g inverse of g of x, since g inverse and g are the same thing. So we also end up with x for this. G here is a special kind of function. It's the inverse of itself.

Reflect the Vertical Line

One important feature of inverse functions that we haven't really talked about, is the fact that they are, well, functions. Here for example, are the functions we were working with in the last quiz. We started out with the equation y equals 4x, and found that its inverse is y equals x over 4. Of course, the original equation is a function, but we can tell that its inverse is too. It passes the vertical line test. We saw that y equals 4 over x is its own inverse. It passes the vertical line test, too. Another fun fact about this, is you can tell that it's its own inverse, because the function itself is symmetric across the line y equals x. Now if we had a function and we went through the process of finding its inverse, but the resulting equation did not graph a function, then the original function would not have an inverse at all. I know that sounds kind of complicated and that would involve a lot of steps. So luckily, there's an easier way to tell whether or not a function will have an inverse. It involves graphing. Now I just said that we can figure out if a graph represents a function by using the vertical line test. And we also know that in order to graph the inverse of a function, we need to reflect it in the line y equals x. So, what if we decided to reflect a vertical line in the line y equals x? Here, I've graphed a random vertical line. I happen to pick x equals 4. And this dotted black line is the line y equals x. What is the reflection of x equals 4 in the line y equals x? Please write the equation of this new line in this box.

Reflect the Vertical Line

The reflection of x equals 4 in the line y equals x is just y equals 4. We can see on the graph that the line y equals x cuts this picture exactly in half. This part of x equals 4 gets mapped over to this part of y equals 4, as does this part to this part.

Number of Intersections

So now we know that lines y equals 4 and x equals 4 are reflections of one another in the line y equals x. If we flip either of them across this line, we get the other one. Now, let's say we have some function that only intersects the line y equals 4 one time. If we find its inverse, how many times will that curve intersect the line x equals 4?

Number of Intersections

Its inverse will only intersect x equals 4 one time. The relationship between the reflection of the function and the line, x equals 4, will be the same as the relationship between the original function and the line y equals 4.

Testing for Inverses

Let's take a second to think about the past couple of quizzes you've done. We saw that a vertical line, reflected in the line y equals x, becomes a horizontal line. Now, just like we have the vertical line test to determine if graphs are functions, remember, it says that a given curve is a function if it only crosses a vertical line at one point for each value of x in the domain. We can perform another test on the graph to figure out if its reflection in the line, y equals x would be a function. That would mean that it has an inverse. Which of these statements do you think is true, then? A function has an inverse if no horizontal line passes through its graph at more than one point. A function has an inverse if no vertical line passes through its graph at more than one point. A function has an inverse if all horizontal lines pass through its graph at more than one point. Or, a function has an inverse if the line, y equals x, passes through its graph at more than one point. Think back to the vertical line test, and how functions are related to their inverses.

Testing for Inverses

This first choice is correct. A function has an inverse if no horizontal line passes through its graph at more than one point. Now if a function passes this test, if it meets this criterion, which we call the horizontal line test, then that means that its reflection in the line y equals x will pass the vertical line test. Which means that, that reflection is a function. If we verify that the reflection of a function in the line y equals x is itself a function, then that means that the original function does in fact, have an inverse.

Do they graph inverses

Let's put that brand new understanding of the horizontal line test into action. Here are 3 functions, we have f of x equals 2x to the 5th, g of x equals 3 times the opposite value of x minus 2, and h of x equals 2x squared. Please tell me whether each of them has an inverse. Remember the horizontal line test says that a function will have an inverse. If any horizontal line we can draw on the coordinate plane will only pass through the original function one time.

Do they graph inverses

The only function of these three that has an inverse is this first one. F of x equals two x to the fifth. Neither this porabala nor this absolute value function pass the horizontal line test. So neither of them has an inverse. This function does, however.

Are they invertible

Maybe there is hope for these functions, though. Perhaps some domain restrictions could help us out. After all, if we only look at part of these graphs, it might be that the part we leave passes the horizontal line test. That will mean that, that part at least, would have an inverse. Let's just focus in on one of these two for the moment. If we look at this function, only in the part of the domain where x is greater than or equal to 0, does that function have an inverse? And then secondly, if we look at the function only where x is less than or equal to 0, does that function have an inverse? Think about which parts of this graph each of these are looking at, and then decide whether or not they each pass the horizontal line test.

Are they invertible

And the answer is, that both of these functions have inverses. Here are their graphs and we can see, that they each pass the horizontal line test. Once again, here's the original function. H of x equals 2x squared with no restrictions on the domain at all. So, even though this function doesn't have an inverse, because it doesn't pass the horizontal line test. And apparently, for some functions at least, if we change the domain of the function, we can influence whether or not it has an inverse. That's pretty neat, domain continues to be super powerful and really, really important to keep in mind.

Restrict the Domain

We saw before that this absolute value function, g of x, equals 3 times the absolute value of x minus 2, does not have an inverse when we consider the entire domain, all the real numbers. But maybe if we apply some restrictions to the domain, we can make this function have an inverse. Considering each of these choices on its own, which of them would make the function g pass the horizontal line test? Please note, that if you pick more than one choice here, that doesn't mean that we have to use both of those parts of the domain at the same time. We're just looking at these one by one.

Restrict the Domain

Here are the portions of the domain out of these choices that, when applied to g of x, would make it have an inverse. If we look only at the portion of the graph where x is less than or equal to 0, that's this part right here. That definitely passes the horizontal line test. If we look only where x is greater than or equal to 3, that's from here, onward, to the right. That also works. For negative 2 is greater than x, we have again, a part of this side of the graph. And for 7 is less than or equal to x, we have part of this branch over here, as well. So this is yet another case where looking at only parts of the domain of a function can make that function have an inverse.

Inverse Finale

Before we wrap up our discussion of inverse functions, let's do a quick recap, and challenge problem. Suppose we have the function, f of x, equals 2x minus 3, over 5. What is its inverse?

Inverse Finale

Starting out with our function, f of x equals 2x minus 3 over 5. I've gone through the same steps that you've become very familiar with during this lesson. First, I switched from f of x to y to make a notation easier to handle. Then, to switch to the inverse function, I interchanged x and y. So instead, we have the function, x equals 2y minus 3 over 5. Then I went through a few stuffs to solves for y. Getting y by itself on one side gave is the equation 5x plus 3 over 2 equals y. Then I just switched the two sides of the equation and changed y to f inverse of x. That gives us a final equation of f inverse of x equals 5x plus 3 over 2. Let's double check that this is the right answer. Remember, we can do that by looking at the composition of f inverse and f and the composition of f and f inverse. f of f inverse affects is 2 times the quantity end up with x. Let's try the composition, the other way around. For f inverse of f of x, we have 5 times the quantity 2x minus 3 over 5, plus 3 all over 2. And sure enough, cleaning this up, we get x as well. We've now verified that f inverse of x really is this expression. 5x plus 3 over 2. I hope at this point you feel really comfortable finding the inverses of different functions. And then checking that they actually are their inverses. Remember this also means that not only is 5x plus 3 over 2 the inverse of 2x minus 3 over 5, but 2x minus 3 over 5 is also the inverse of this function. They're inverses of one another. It's a two-way relationship.

Properties of Inverse Functions

To round things out, let's double-check the subtleties of your understanding of inverse functions. I've written a bunch of statements here and I'd like you to tell me which ones are true. Now, we may not have talked explicitly about every single one of these. So, think about each of them carefully, and if you need to try more than once, that is completely fine, as always. You can check off as man as you want to, even all of them, if you think that that's appropriate. So here are your choices. All functions have inverses. All inverse functions have inverses. A functions domain can influence whether or not it has an inverse. All functions pass the vertical and horizontal line tests. And lastly, all inverse functions pass the vertical and horizontal line tests.

Properties of Inverse Functions

Let's go through each of these statements to decide whether or not they're each true. So, first things first, all functions have inverses. Well we know that that's not true. Some functions, like y equals x squared, don't pass the horizontal line test. So, this doesn't have an inverse. We can't check this first one off then. Moving onto the second one, we have all inverse functions have inverses. This one is true. If we start off with one function, and reflect it across the line y equals x. If we start off with one function, and then reflect it across the line, y equals x and end up with another function, then the second function is the inverse of the first one. But if we reflect the second one back across the line y equals x again, we'll end up with its inverse, which is the first one. This is basically the same as the property I showed you before. If we have a function f inverse that we know is the inverse of f, and then we take its inverse, that just equals the original function f. So yes to the second choice. Now third, we have a function's domain can influence whether or not it has an inverse. This one is definitely true. We saw this with our parabola, which we took only portions of, and saw that those portions had inverses, while the whole parabola didn't. And we also saw the same thing with our absolute value function. Fourth, all functions pass the vertical and horizontal line tests. Well all functions pass the vertical line tests. That's how we know that they're functions. But not often past the horizontal line test. So this isn't true. The last choice however, is true. All inverse functions pass both the vertical and horizontal line tests. Inverse functions are functions, so they pass the vertical line test, and since inverse functions all have inverses as we've already decided, that means they have to cross the horizontal line test, as well. Now I personally think this was a really, really tough quiz. This required you to show some very in-depth understanding of inverse functions, and to think really critically about all of these statements. Awesome job on this quiz and on the entire lesson.