 Let's see, this is episode seven of our Math 1050 College Algebra class. I'm Dennis Allison, and I teach mathematics here at UVSC. Today, we'll be talking about one-to-one functions and inverse functions. Now, what I mean by an inverse function is a function that will undo what a function did, what a function might do. Now, not every function has an inverse function, so we want to decide, first of all, which ones have inverses, and then what those inverse functions are. We'll talk about something called the inverse function property, and then we'll find two ways of finding the inverse of a function. And finally, we'll consider the graph of an inverse function when the inverse function exists. Now, just to start things off, let me remind you of a few ideas about functions. You remember that a function is an association between two sets. There's the domain and the range. And the function has the property that each member of the domain is assigned to exactly one member of the range. Although sometimes, you can't have several members of the domain assigned to the same range member. Let me give you an example. Suppose we take the function f of x equals x squared. Now, it looks like if we don't restrict the domain, that the domain of this function is all real numbers, because I can plug in any number here for x and I can square it. And what does it look like the range is going to be for this function? It's all numbers greater or equal to 0. It'd be all numbers 0 or larger. So I'm going to write that as the closed interval from 0 to plus infinity. But you notice that if I substitute in for x, if I substitute in 2, f of 2 is 4, and if I substitute in negative 2, f of negative 2 is 4, so what that means is that if I draw my function illustration, let's say this is the domain of the function f, and this is the range of the function f, then I have the number, let's say this is 2 right here, associated with 4 by way of f. And then if this dot represents the number negative 2, it's also associated with 4, so I have two different x's that are associated with the same y. Now, for this reason, I would say that this function is not a one-to-one function. What we mean by a one-to-one function is this. Let me write the definition here on the screen. A function, we'll call it f, is one-to-one if f of x1 equals f of x2 only when x1 equals x2. And the function we just looked at didn't have this property. You see, f of negative 2 was equal to f of 2 when I was discussing, when I was looking at the function f of x equals x squared. But the problem is that negative 2 does not equal 2. So in order to be a one-to-one function, whenever the two function values are the same, then the two numbers that we've substituted in would have had to be the same. And in this case, I was able to find two different numbers that gave me the same function value. Let's take a look at a couple other functions and see if we can decide if they're one-to-one functions or not. Let's take, for example, the function I'll call this one g. g of x equals 2x plus 1. By the way, if I graph that, what would it look like? A straight line. That would be a straight line. Exactly, it crosses the y-axis at 1, and it has slope 2. Now, suppose that I have two numbers, let's say x1 and x2. And what if I say that g of x1 is equal to g of x2? And so I ask the question, is it necessarily the case that x1 equals x2? Let me put a question mark over that, because we don't know yet if that's true or not. Well, let's see. If g of x1 equals g of x2, then let's see. If g of x1, if I put in an x1 right here for x, this would be 2 times x1 plus 1. 2 times x1 plus 1. And g of x2 would be 2 times x2 plus 1. So these two expressions would have to be equal. If I subtract 1 from both sides, then I have 2x1 equals 2x2. And if I divide both sides by 2, then x1 does equal x2. So if x1 and x2 have to be equal, what this implies is the g is a, and here I'm going to use an abbreviation, is a 1 to 1 function. So if you just put 1 dash 1, I'll know that that means it's a 1 to 1 function. Let's take another example. What if we take the function, I'll call this one capital F of x, capital F of x equals the square root of x minus 3? You know, if I graph this one, we haven't seen this one for a couple episodes, but this is the fundamental square root function with a shift on it. We have to move it three units to the right. So if I move it three units to the right, so here's three, then there's a target point at the new origin. And if I go over 1 and up 1, let's say that's going up 1 right there. So those are the two target points. And what I get is half of a parabola. So this graph looks like that. Now I'd like to know if this is a 1 to 1 function. So I'm going to say, suppose F at x1 equals F at x2. Does it follow that x1 and x2 are in fact the same number? Well, F of x1 would be the square root of x1 minus 3, and F of x2 would be the square root of x2 minus 3. So to find out if x1 and x2 are equal, I'm going to square both sides to get rid of the radicals. Well, when I square the left-hand side, I have x1 minus 3. When I square the right-hand side, I have x2 minus 3. And then if I add 3 to both sides, x1 is equal to x2. And therefore, this tells me that the original function F is a 1 to 1 function. So that says if I were to pick two different numbers in the domain, I would get two different function values in the range. OK, and for my last example, suppose we take the function, let's say h of x equals x squared minus 4. Now, what would you guess, class, for this function? Do you think it's going to be a 1 to 1 function or not? Ginny says no, Stephen says no. Well, let's just see. Suppose I say that h of x1 equals h of x2. So we're starting off in the same fashion that we did for the other examples there. h of x1 could be replaced by what? What could I substitute for h of x1 using that definition? x1 squared minus 4. x1 squared minus 4, OK. And in place of h of x2, I'll put x2 squared minus 4. We could add 4 to both sides. And I get x1 squared equals x2 squared. And then if I take the square roots on both sides, x1 and x2 don't have to be equal. What I could say is that x1 is equal to either plus or minus x2. Either way, their squares would be the same. Well, because it's not necessary that x1 is actually equals x2, but it could be the negative of x2, what this tells me is my function h is not a 1 to 1 function. So this one is not a 1 to 1 function. OK, well, this is all very abstract. Let me see if I can show you some other ways of looking at what a 1 to 1 function would be. Suppose I draw an illustration like this. Here's the domain of a function. Let's say we'll call this function f. And over here is the range of a function f. And I'm just going to pick several numbers to substitute in here for the domain. Let's go to the next graphic. And I think you can see how we would choose these. Suppose in the domain a, I have the numbers 1, 2, 3, and 4. You see that in the illustration on the left. And those numbers are mapped to 2, 4, 7, and 10 in that order. So 1 is, as we say, mapped to 2. 2 is mapped to 4. 3 is mapped to 7. And 4 is mapped to 10. I don't know that I had any particular rule in mind when I made those assignments for this graphic, but let's just say that those are the domain and the range members. You notice that if I pick two different members of the domain like, say, 2 and 3, then they are associated with two different members of the range. So in the illustration on the left, f is a 1 to 1 function. Now if you look at the illustration on the right, again in the domain, we have 1, 2, 3, and 4. And in the range, we have only 2, 4, and 10. And I don't believe that if I pick two different members in the domain, I have to get two different associations in the range. In that illustration, do you see two numbers in the domain that map to the same member in the range? Yeah, which ones, David? 4 and 3 both go to 10. Let's see, let's look at that again. Actually, it's not 1 and 3 going to 10. It's 4 and 3. Oh, 4 and 3. Yeah, 4 and 3. I just misunderstood you. 4 and 3, each mapped to 10. So if I draw that on the board, let's say I have 1, 2, 3, 4. And over here, I have 2, 4, and 10. 1, 2, 3, 4, 2, 4, and 10. What we turn our attention to is this relationship. 3 is mapped to 10, and 4 is mapped to 10 by way of F. I'll just write F on the arrow, because the arrow is actually making the association. So we could say that F of 4 is 10. And we could say that F of 3 is 10. And therefore, F of 4 equals F at 3. But 4, of course, does not equal 3. So just as in the previous examples that we showed, we have a situation now where I have the same function value for two different x's. And what that tells me is this function F, actually the graphic was called g. But this function F is not 1 to 1. OK, now if there is an inverse function, then there is a certain property that's satisfied called the inverse function property. And let's go to the next graphic after that 1 to 1 function graphic. OK, so here we have the formal definition of what is a 1 to 1 function. And it says that a function F is a 1 to 1 function if no two members of the domain are associated with the same member of the range. Now, 1 to 1 functions become quite important when we go to inverse functions. So let's just go to a graphic. And we'll see a little bit about inverses. You notice on the left we have a domain set A, range set B. This is the domain of set A. This is the range of function F. This is the range of function F. So I start off with a number X in the domain. And F sends it over to F of X. And then F inverse sends that number back where we started. And so if I label these as the domain of F and the range of F, then what we can say is if you start off with X and if you take F of that so that you're now in the range of F, and then you send it back using F inverse of F of X, then you come right back where you started F of X. So this says F sent the X away, F inverse sends that back, and we're right back where we started. So we've made a round trip. And the whole notion of an inverse function is that it returns it where it started. Now, if you look at the illustration over here on the right-hand side, then this is the domain of F inverse. And this other set is the range of F inverse, because F inverse is going the other way. Now, you see, we start off with the number X, which is in the domain of F inverse. F inverse sends it away to its range. So here's F inverse of X. And now F sends it backward started. So we have the same circular path, but we're starting from the other set. This says if I start off with X, and if I take F inverse of that, and then I take F of that, we're right back where we started with X. Now, if we come back to the green screen, let me just summarize this information with what's referred to as the inverse function property. And it says if F has an inverse function, F inverse, then the following two statements are true. If I take F of X, and then if I take F inverse of that, I get X. On the other hand, if I start off with an X, and I take F inverse of that X, and then take F of that, I'm back to X again. Now, there's another way that I can write this composition of functions. If I take F inverse of F of X, you remember a composition can be written as F inverse circle dot F of X. Now, you see, when I write this, this is a single function. And it tells me the history of the function. It came from composing F and then F inverse. On the other hand, the composition that I have on the second line says I start off with X, and I have the composition. Let's see, F would go here composed with F inverse. So here's the composition in the reverse order. And each time that I apply that to the variable X, I come up with X again. In the last episode, we talked about compositions of functions. And we said that generally, when you reverse the order of a composition, you don't get the same answer. But for inverse functions, this is one of the rare times that you do, because you end up taking X, and you send it right back to X using either inverse composition. OK, let's see, if we go to the next graphic, it says how can we go about finding an inverse function? Well, first of all, one way to find out whether I have an inverse function is I can use the composition of two functions. Given two functions, F and G, we can decide if they are inverse functions by taking their composition. And so if I take F composed with G of X, I should get X. And if I take G composed with F of X, I should get X. OK, here's an example of what that idea is about. Suppose I have two functions. F of X is X plus 3, and G of X is X minus 3. Now, we might ask ourselves the question, is the function G of X the inverse of the function F? In fact, does F even have an inverse function? Well, one way to find out if these are inverse functions is to take their composition, take F composed with G of X. And if G really is the inverse of F, then this should equal X. It should equal X. Let me just put a little X over there. That's what I should arrive at. Now, let's see, how can I get from here to there? In fact, if that is the answer. Well, I'm going to take F of G of X. And then I'll take F of, and I'm going to substitute for G of X, X minus 3. And then I'll take F of that. Now, let's see, what does F do to anything? Well, F takes any number, and it adds 3 to it. For example, what would be F of 5 in this function? 8. It'd be 8. What would be F of 0? 3. 3. What would be F of A? Suppose A is a number. What's F of A? A plus 3. A plus 3. What is F of X minus 3? X minus 3? X. Plus 3. It's going to be X minus 3 plus 3. So in other words, I'm doing what the function said to do, and that's to add 3 to whatever number that I began with. And this reduces to be X. And by golly, we got exactly what we said we should get. So this indicates that G must be the inverse of F. Let's try this in the reverse order and see if the same thing happens. This time, I'm going to take G composed with F of X. Because if you remember, if this is the inverse function, I should be able to compose these in either order, and I should get X when I'm done. Now, the composition of these two functions means, literally, that I take G of F of X. That's what the composition means. And this is equal to G of, let's see now, inside. I'll substitute for F of X. That's X plus 3. And what does G do to any number X? Well, it subtracts 3 from it. So what does G do to this? It's going to subtract 3 from it. X plus 3, I'll subtract 3 from it. And by golly, I end up with X. So these two calculations, this one and the one I did just before, indicate that F and G are inverse functions of each other. They're inverse functions of each other. So in other words, if I use the space right here, G is actually the inverse of F. And by the same argument, F is the inverse of G. Yeah, Ginny? Can you have one that's an inverse of one, but not be the inverse of another? No, if one's the inverse of the other, then the other will be the inverse of the first one as well. So each one undoes what the other one did. If you look at this from a little bit different point of view, what F does is it takes the number X and it adds 3. Now, how do you undo adding a 3? Well, what you do is you subtract a 3. So what G does is to subtract the 3 back off. So if you add a 3 and then you apply G, which says subtract a 3, you should be right back where you started. So these are inverse functions of one another. Let me give you sort of a silly analogy to this. But what would be the inverse function for tying your shoe? You untie your shoe. So you start off with your shoe untied. You tie your shoe. The inverse function is to untie your shoe and you're right back where you started. Your shoe's untied. What's the inverse function for putting on your socks? Taking off your socks. Taking off your socks. OK, then you're right back where you started. You don't have any socks on. What's the inverse function for shooting somebody dead? Well, there's no inverse function for that. You can say, gee, I'm sorry. I didn't mean to do that. Could you get up now? But there's no inverse function for that. And there are functions that have no inverses, but the ones that we've just seen, this f of x and g of x, were inverses of each other. So we don't want to think that every function has an inverse, just like in life. Not everything we do can be undone. But if we can undo things, that gives us new power in what our options might be. Let's take another example. This time, I'm going to choose two functions a little bit more complicated. f of x is going to be, let's say, 3x minus 4. And g of x is going to be x plus 4 over 3. These look like they're somewhat similar. I mean, this is a linear function. And the one over here is a linear function. You might say, wait a minute. That doesn't look like a linear function. Well, if I actually divided 3 into that, I could say that's 1 third x plus 4 thirds. Now it looks like a linear function. y intercept 4 thirds and slope 1 third. Are these inverse functions of each other? I'm going to go back with my original statement for what g of x is. Well, what I'll do is take the composition. f composed with g of x. And then I'll also work out g composed with f of x. Let's see. Now, literally, g dot f, the composition of f and g, excuse me, that's f composition g, is f of g of x. Now, do you remember what was my first step when about taking this composition? What's the first thing I do to reduce this? Put in f of, and then put the x plus 4 over the 3 in it. Right, right. We're going to work from the inside out. I'm going to substitute for g of x. And this would be f of x plus 4 over 3. OK, and now I want to find f of that. Well, let's see. Now, what does f do to anything? No matter what you put in for x, it's going to triple it, and then it's going to subtract 4. So I'm going to triple this, and then subtract 4. So I'm going to triple x plus 4 over 3, and then subtract 4. Well, these 3's cancel off. And so that's going to be x plus 4 minus 4. And that reduces, of course, to be x. So that tells me that these two functions should be inverses. And I'm thinking if I reverse the order of the composition, I should get the same answer. I'll just try that. We'll work through that fairly quickly. I'm going to reverse the order and see if we get x in that case. So g composed with f of x is g parentheses, f of x, which is g parentheses. Let's see now, what should I put in there? 3x minus 4. Yeah, because we're substituting for f of x. 3x minus 4. And now what does g do to this? Well, g takes any number, it adds 4 to it, and it divides by 3 afterwards. So this is going to be 3x minus 4. I'm going to add 4 to it, and then I'm going to subtract 3. You see, if I change what goes here, I have to change what goes there. And I put 3x minus 4 here instead of x. So I put 3x minus 4 here instead of x. And this gives me 3x over 3, and that gives me x. Now, you notice what's happening is I'm getting the same result, but I'm going through a different series of steps. This doesn't look anything like the calculation I just erased. This is a ratio that reduces to be x. The other one was a sum and a difference, I think, that reduced to be x. So I go through a different process, but I arrive at the same result. That's because I reverse the order of the functions. And because I get x in either order, once again, these two functions are inverses of each other. So if I were to ask you, given function f, what is f inverse of x? What's f inverse of that function? Well, it's this one over here. So the inverse function is x plus 4 over 3, or if you prefer to write it, it's 1 third x plus 4. Now, what if looking at, oh, should be? Oh, 4 thirds. Yes, thank you very much, 4 thirds. Now, if we were to start with g of x, what is g inverse of x for this function? 3x minus 4. It's 3x minus 4, exactly. So we know that these are inverses of one another. Now, on your homework, you're going to find some problems where they will give you two functions, a function f and a function g, and they'll ask you, are these inverse functions of each other? And the way you do it is you take their composition in either order and see if when you start with x, you end up with x. OK, now this x represents an arbitrary member of the domain. If I plugged in a 3 there, I should get a 3 here. If I plug in a 0 there, I should get a 0 here. But rather than doing particular values, I'll pick sort of a generic variable, x, and I come back with x. OK, well, let me ask you this. Are these two functions inverses of one another? This function says f of x is the square root of x plus 2. And this function, if it's the inverse, let's see. Now, that was a square root. I'm going to put a square on this one. I'm going to say x plus 2 squared. That looks like that could be a reasonable candidate for an inverse here, because this had a square root. So this has a square. That's the opposite of what I did there. The way I would find out if these are inverse functions is I'd take their composition. I'll take f composed with g of x. And that's f of g of x. And that's going to be f of x plus 2 squared. And that's going to be, well, let's see. We have to be careful now. What does f do? It takes a number. It adds 2 to it. And then I take the square root of the whole thing. So I'm going to take the quantity x plus 2 squared. I'm going to add 2 to it. And I'm going to take the square root of the whole thing. Do you see that this is going to, do you see any way this is going to reduce to be x? I don't think so. I don't think there's any hope of us getting an x out of that. So what this means is these are not inverse functions of each other. We may have thought they might be, but they're not. But I bet there's a variation of this that would be the inverse. You might say, well, Dennis, what is the variation of it? And how would you know? How are you coming up with these functions to ask us about? Well, that actually leads us to the next idea. Given a function, how can I figure out what its inverse function is? So if we go to the next graphic, I'm going to show you sort of a step-by-step procedure that will allow us to find the inverse of a function if it has an inverse. This is called the algebraic method. Given the rule of a one-to-one function f, the rule of f inverse is found by, first of all, replacing f of x with y. And then solving for x, and then interchanging the x and the y, and then inserting f inverse of x for y. And there's a note of warning here at the very end. It says, be careful to restrict the domain of f inverse when necessary. And we'll have an example where we need to restrict the domain in a moment. Now, you notice in that opening statement, it says, given the rule of a one-to-one function f, you can use the following procedure. Well, this brings up the connection between a one-to-one function and an inverse function. The functions that are one-to-one functions are precisely those functions that have inverses. Let me just discuss that, and then we'll look at the algebraic method. If I go back to the notion of a one-to-one function, you remember, if this is the domain of a function and this is the range of a function, suppose I choose a number 3, and I map it over to 10, and then here I choose the number 4, and I send it over to 10. I'm going to call this function f. On the basis of that illustration, is this a one-to-one function? No, it's not. This is not a one-to-one function. And look what happens if I try to send 3 to 10, and then I want to send it back where it came from. When I send it back, when I send 10 back, I could just as easily send 10 back to 4 and not back to 3. So in order to avoid the complication of sending 3 to 10 and then returning 10 to 4, not to 3, what I do is make sure I have only one number in the domain that goes to 10. So then when I return it, I have only one place to return it to. So that means if this had been a one-to-one function, I could have produced an inverse for it. And one-to-one functions and functions with inverses are precisely the same thing. OK, now that brings us to this algebraic test. I'm going to take a function like f of x equals, let's say, we have 2 minus 3x. And I want to find out if this function has an inverse or not. Well, the first question I might ask is, is this a one-to-one function? If it's a one-to-one function, it has an inverse. Now earlier in this episode, what I did is I tried taking f of x1, and I tried taking f at x2, and then I asked the question, does that imply that x1 equals x2? Well, I could go through those very same steps again, and I think I would find out that x1 and x2 have to be equal, but let me show you a little bit quicker way of finding out if this is a one-to-one function. I'm going to draw the graph of this function. I'll just put the graph over here. And let's see. This is the same thing as saying negative 3x plus 2. So it's a linear function. It crosses the y-axis at 2. So it crosses the y-axis at 2. And the slope is negative 3. So if m is negative 3, that means negative 3 over 1. So if I go over 1, I should go down 3 and get another point right here. So when I draw the graph of f, graph of f looks, well, something like that. Now you notice this function has the property that if I draw horizontal lines through the graph, horizontal lines intersect at only once. That's what I'll call the horizontal line test. And if a graph passes the horizontal line test, it will be a one-to-one function. So a horizontal line crosses only once there. A horizontal line crosses the graph only once there. A horizontal line crosses the graph only once there. Let me just show you a function over here on the side that wouldn't have passed that property, the horizontal line test. If I draw a parabola, for example, I won't bother plotting the target points here. You notice this time a horizontal line can cross it two times. Let's say this is the function f of x equals x squared. x squared. And let's say this is the horizontal line that crosses at four. Well, there is a number right here, two, that makes this the ordered pair, two, four. And there's an ordered pair over here at negative two. So that this is the ordered pair, negative two, four. You remember, we talked about the function f of x equals x squared earlier at the beginning of this episode. And this function was not a one-to-one function. And here's why. If I draw a horizontal line through it, say at four, I can find two different x's that have the same y value, and therefore they're both on the horizontal line. So if a horizontal line crosses a graph more than once, it's not a one-to-one function. Okay, that's the horizontal line test. Well, the function that we're considering passed the horizontal line test. So I'm gonna say, according to the horizontal line test, f is a one-to-one function. I'll use my abbreviation for one-to-one. So this is a one-to-one function, and therefore it should have an inverse. Okay, so what is the inverse function for it? Well, I'm gonna calculate it right below here. Okay, this algebraic test says, given the rule of a one-to-one function f, the rule of f inverse is found by following this four-step procedure. First of all, you replace the f of x with the variable y. Then you solve for x, and then you interchange the x and the y, and then finally you insert f inverse of x for y. Now, let me explain how that works and why it works along the way. So we have our function here, f of x equals two x minus three, and the horizontal line test told me that according to the graph, this is a one-to-one function, and therefore it has an inverse. So my first step is I'm gonna replace f of x with y equals two minus three x. Because y and f of x are essentially equivalent, they're just different names for the second, for the second coordinate of the function value. So I wanna now solve for x. You know, at the moment I would say this equation has been solved for y because the y has been isolated. So now what I wanna do is to solve for x, and that is to isolate the x. So I think I'll subtract two from both sides. y minus two equals negative three x. I'd like to get my x's on the left, so what I'll do here is just flip the whole thing over, and I'll put the negative three x equals y minus two. Then I'll divide by negative three. So x is equal to y minus two over negative three. And just to make this look a little bit simpler, I'm gonna multiply the top and bottom by negative one, so that'll make that a positive three. And when I multiply by negative one on top, that makes the two positives, I'll put it first. That's a two minus y. Let me make my two a little better. It looks sort of like a z there for a moment. Okay, and so now I've isolated the x. The next thing I do is to interchange the x and the y. Now you might say, wait a minute, what's that all about? Well, you see the idea is I wanna change the notion of what's the domain and what's the range. This x came from the domain of f, and the y came from the range of f. But I want this to become the range of f inverse, and I want this to come from the domain of f inverse. That is my sets a and b in my earlier illustrations are now playing different roles. This y comes from the domain of f inverse, so I'm gonna call it x. And this x comes from the range of f inverse, so I'm gonna call it y. Well, when I make that interchange of x and y, I'm now talking about the inverse function, not the function f. So when I eliminate the y, I won't call it f of x, I'll call it f inverse of x. And this is equal to two minus x over three. If you prefer, you could just as easily call it two thirds minus one third x, but somehow this looks a little bit more compact for my purposes. So the inverse of the function f that I have here is two minus x over three. How could I go about checking this independently to see if this is the inverse of that function? What did we do earlier to see if two functions are really inverses? You did the composition of it. Yeah, we did the composition, we used that inverse function property. So I'm gonna take the composition of these two to verify that that really is the inverse of this one, and then we'll do another example or two of this process. The composition, well, what if I take f inverse composed with f of x? Okay, now this is just like what we did just a few minutes ago. This means I'm gonna take f inverse of f of x, and I'm gonna substitute for f of x, f inverse of two minus three x. And what does f inverse do? Well, it takes two minus the number x and divides by three. Well, I don't have x, I have two minus three x, but this will be two minus the quantity, two minus three x divided by three. Now, if I distribute the negative, that says two minus two plus three x. All over three, yeah. What happened to that three in front of that x? The three in front of the x, the three right here, the three, we've got it. Oh, okay. It's there. Okay, I'm sorry. So if I reduce this, the twos cancel, and I have three x over three, and that reduces to be x. Yeah, that's exactly what we were expecting to get, because we think this really is the inverse function. We just calculated that by the algebraic method. And rather than calculating the composition in the reverse order, I think you know how to do that, so I'm gonna skip that, but you'd find out that if you took f composed of f inverse of x, you'd end up with x also. Rather than doing that, let's do another example to see how I calculated my inverse function. Okay, this time, I'm gonna take the function capital F of t, just to pick another variable, I'm gonna say t, and this is going to be t minus two squared plus one. Now, first of all, the question might be, is this a one-to-one function? Does it even have an inverse? So I'm gonna come over here and graph it, and I'm gonna have to call my horizontal axis the t-axis, I'll call this one the y-axis as before. And if you remember, this is a quadratic function, it's basically a transformation of f of t equals t squared. What transformations are made in the fundamental graph? Right two and up one. Yeah, so I've gotta go to the right two, and I have to go up one, and then I'll graph that fundamental parabola. So I'm gonna go to the right two, here's two, I'm gonna go up one, and so the vertex is right there. Now there are two other target points, and there's no stretch on this function because the coefficient is one. So if I go over and up one, I get a point there, and if I go back one and up one, I get a point here, and so I get a parabola that looks like, well, looks sort of like that anyway. But this is not a one-to-one function, it doesn't pass the horizontal line test. I can make it a one-to-one function if I put a restriction on the domain. What if I say I only wanna consider the portion of the graph to the right of that dotted line? I'm gonna throw this away. I'm gonna throw that away. Now you might say, well gee, can you just throw half of it away? Well I can if I say here, I want my t's to be greater than or equal to two. That's a restriction on the domain, and now I have only this half of the graph, I have this restriction on the domain, and the domain is not the largest possible set of numbers that I could substitute in, but I forced it to be a limited set. Now would you say that the graph that we now have, does it pass the horizontal line test? Yes, it does. It does pass the horizontal line test. So that means there must be an inverse function. So the question we ask is, what is F inverse? That is what is the rule for F inverse? So to calculate that, the first thing I do is I replace F of t with y, and over here on the side I'm gonna make mention, just mention the fact that t, remember is greater than or equal to two. Then the idea is to solve for, well not solve for x, in this case I'll solve for t. So what would you do first if you're trying to isolate the t? Subtract a one from each side. Subtract a one from both sides, y minus one is t minus two squared, and remember t here is greater than or equal to two. Okay, then I'm gonna take a square root on both sides. But this presents a bit of a problem, because when I take the square root of y minus one, there's the positive or negative square root, technically. And so which square root should I take? Well, you remember the restriction that t is greater than or equal to two? So what can you tell me about t minus two? What is it greater than or equal to? Zero. It's gotta be at least zero, because t is at least two or larger. So this square root couldn't be negative, because this quantity is zero or larger, so I know to take the positive square root only over here. I don't think I'll bother writing that plus, we understand this to mean the positive square root. And you see that's exactly why we restricted the domain over here, so that when I get to this step, I know which square root to take. So now I can continue to solve for t. If I put the t on the left, that t is gonna equal this square root plus two. The square root of y minus one plus two. Okay, and once again, I remember that t is at least two or larger. You know, that makes sense even now, because this square root, being a positive square root, has to be at least zero, and when I add two to that, t has to be at least two, just like what we've been saying over here all along. Okay, now I've isolated the t, now the next step was to interchange the two variables. So I'm gonna put y here, and I'm gonna put the square root of t minus one plus two. Okay, now the restriction that t greater than or equal to two changes to y greater than or equal to two. The y over here is greater than or equal to two. So when I reverse those variables, I have to reverse the y, reverse the t over here. And when I reverse my two variables, I'm now talking about the inverse function. So this is f inverse of t is the square root of t minus one plus two. Let's go ahead and graph the inverse function in this illustration over here. This is the square root function with two changes on it. You remember a square root function has a graph that sort of looks like that? So what changes will I make in this graph? To the right one and up two. Need to shift it to the right one. Need to shift it up too. If I go to the right one, up two, I get a target point right there. And then there's another target point if I go over one and up one right here. Let me move that f up out of the way. And this square root function looks like that. And this is f inverse. So if I draw them on the same coordinate system, I notice those two graphs look a lot alike. As a matter of fact, one is the mirror image of the other. And this is something we're gonna talk about more in some detail in a moment. But if I draw a 45 degree line right through the origin, right through here, this is the line y equals x that I'm graphing there. If I take the original graph of f and if I flip it over that 45 degree line, and if I take this portion of the graph of f and if I flip it over to the other side, I will see the graph of the inverse function. Okay, this is a common characteristic that we're gonna see now in some more examples. I'd like to do yet one more example of calculating the inverse of a function using this algebraic method right here. And I wanna pick one that's a little bit more complicated like you will, you'll see all of these examples like all of these in your homework, but I wanna pick one that makes the algebra a little bit more complicated. This one I'll call little g of x equals x plus one over x minus three. Now, you know, we don't know how to graph this function. This is something we've never grap before, so we won't be able to draw an illustration over here on the side like we have in the previous examples. However, let's assume that this is a one-to-one function. It actually is, but I can't verify it with a graph. And the first thing I'll do is replace the y with the g of x with a y. And you remember what the next step was? It was to solve for x. The problem is I have x's in two different places, so how am I gonna solve for x? Well, I'm gonna multiply both sides by x minus three so that I can at least get rid of this rational expression. So I'm gonna take y times x minus three. And if I multiply on the right-hand side by x minus three, what will I get? X plus one. X plus one, yeah, because the x minus three is canceled. So this says xy minus three y equals x plus one. Now, my whole goal here is to isolate the x's, and I have x's on both sides. So what I'll do is get them all together. I'm gonna get all my x's on the left. I think I'll subtract off that x here and here. And I'll move the negative three y to the other side. I'll add three y to both sides, and I get three y plus one. Now my x's have been at least isolated on one side. The next thing I'll do is factor out the x. And so if I isolate the x, it'll be three y plus one over y minus one. The x is isolated. So when you see x's in the top and the bottom, it's just a little bit more algebra, but I can still isolate the x, at least I could in this case. And that tells me that g inverse of x would be, oh, let's say I better, excuse me, first of all I better interchange my variables. y is three x plus one over x minus one. And once I've interchanged the variables, this is now referring to the inverse function, not to the original function. So this says g inverse of x is three x plus one over x minus one. That's the inverse function for the function that I started with. Okay, the algebra here is a bit more complicated, I think you'd agree. But the idea is still the same. If I were to take the composition of these two functions, I should get x. Let me just work, let me just take that composition in one order and let's just verify that that's true, because the algebra there is a bit more complicated as well. So here's g, here's g inverse. I wanna take the composition of g inverse composed with g of x. Actually, I think I better move this up to give me enough room to work this. And I'll move this guy up too. G inverse of x was three x plus one over x minus one. I think that's what I just had up there. So if I take that composition, g inverse composed with g of x, then that's g inverse of g of x. And that's g inverse of x plus one over x minus three. And that's, let's see now, here's where things get a little tricky. If I plug in a number x here, I have to replace it in the numerator and denominator. So when I have this rational expression, I have to put that in the numerator and in the denominator. That's gonna be three times the quantity x plus one over x minus three plus one over x plus one over x minus three minus one. So I think what I'll do to simplify this is multiply on the top and the bottom by x minus three. Cause that allows me to get rid of those denominators. So when I multiply here, I get three times x plus one plus the quantity when I multiply here, x minus three. And when I multiply x minus three on the bottom, I get x plus one and then times one minus x minus three. Okay, that looks a little bit better. Now we just have to reduce it. That's gonna be three x plus three plus x minus three over x plus one minus x plus three. When I reduce the numerator, what do I get? It's four x. We get four x. And when I reduce the denominator, what do I get? Four. Four. By golly, we get x. Just as we were expecting we would get. And if I reverse the order of the composition, I'll still get x again there. Now you might say, Dennis, when you give us a g of x and we calculate the g inverse using the algebraic method, do we have to check it this way every single time? No, not at all. But if you're taking a test, let's say, and you've calculated an inverse function, if you want an independent way of determining if that really is the inverse function or not, then it's the composition that you would go to. Okay, the last thing I wanna look at, and let's go to the last graphic to help support this, is to look at another way of looking at an inverse function. Okay, this graphic says that the so-called graphical method indicates that when given the graph of a one-to-one function, we can draw the graph of the inverse function merely by reflecting the graph of f across the diagonal line y equals x. Now, let me show you what that means. Suppose I have a function and the graph of the function looks like this. Okay, I'll call that function f. Now, it looks like this is a one-to-one function because it passes the horizontal line test, I think. But I don't have a clue as to what the rule of this is. I just drew a random curve up here. But if I can see the graph of the function and if it has an inverse, I can sketch the graph of the inverse function this way. You notice the function f crosses right about here on the y-axis. I haven't scaled this. I don't know what number it is, but that says the inverse function should cross right about here. I'm just reflecting this across the line y equals x. I better draw the line y equals x in right here. Here's the line y equals x. So I'm just looking at the mirror image of that point across this line. Same thing here. It looks like the function f crosses the negative x-axis right there. So if I reflect that across the graph, across the diagonal line, I think it should cross the y-axis right about here. Okay? While we're at it, if I pick a point right here on the function f, if I reflect that across the line y equals x, I get a point right about there. And if I choose this point right here, that's where my graph seems to start if I reflect that across the line y equals x, I get a dot there. Now, if I just connect this point, that point, that point, that point, and you know what, even this point, that's where f crosses the diagonal line y equals x. Now, in this case, if I reflect it across y equals x, it stays right where it is. So I'm just gonna connect all these points and I think I'll make it a little bolder so that it stands out from the other graph. And let's see, I better turn this and it's gonna go up this way. Now, what I'm graphing here is the inverse function of, for the original function f. I don't know the rule for f and I couldn't tell you the rule for f inverse. But if I see the graph of f, I can draw the graph of f inverse. Now, if I know the rule for f, I should be able to calculate the rule for f inverse using the algebraic method that we just looked at a moment ago. So there's the algebraic approach when you're given the rule and there's the graphical approach when you're given the graph. And so this should be the graph of the inverse function. Let me take one more illustration of this idea. By the way, it looks like that goes vertical right there but it shouldn't be, it should be just slightly slanted. So if it goes vertical, of course that wouldn't be a function. Let's pick an example where this is a bit more precise. This time I'm gonna pick a function that's sort of a collection of straight line portions. I'm gonna mark off a scale and let's say that I have a 45 degree line coming into the origin. So this would be like negative three, negative three right here. And then it has slope one half over to here. This would be the point two one. And then it has slope, let's say slope three. One, go up one, two, three right here. So that's supposed to be straight. And this would be the point three, four. And I'm just manufacturing this function with no particular rule in mind. I wanna see what the inverse function looks like. Well, if I draw in the line y equals x, okay, there's the line y equals x right there. Then when I reflect these things over, the point three four becomes the point four three. The point two one becomes the point one two. And the points all along this branch stay right where they are because they're right in the crease of the fold. So here is the inverse function right along there. So this is f inverse. It goes through four three, one two, down to the origin and then along this crease. So this is the inverse function. Okay, well to kind of summarize what we've looked at today is we had the overall goal of discussing inverse functions and which functions have inverses. But in order to do that, we began looking at one-to-one functions and we found out that one-to-one functions are precisely those functions that have inverse functions. There's a horizontal line test that I can use on the graph of a function to see if it's a one-to-one function. And that says that if a horizontal line crosses a graph at most once, then that is a one-to-one function. Then we looked at the consequence of that and that is that one-to-one functions have inverses and I looked at two methods for finding the inverse. There's the algebraic method when I'm given the rule and there's the graphical method when I'm given the graph of the original function. So I can find the rule and I can draw the graph of the inverse function based on what's given to me. And you notice basically I can give back the sort of information that I'm given. If I'm given a rule, I can calculate a rule. If I'm given a graph, I can calculate a graph. Inverse functions will play a very important role later in this course and in future courses as well. Let's see, this is the last material that's on the first exam. Next time we get together, we'll review for the exam and then in episode nine, we'll move on to some new material. I'll see you next time.