 Welcome back to our lecture series Math 4220, Abstract Algebra for Students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Misseldine. This is the first video for lecture three in our series where we're going to talk about inverse functions, how they're related to bijective functions we defined in the previous lecture. We'll also talk about some permutations near the end of this lecture as well. Now, in terms of content, a lot of this is still coming from section 1.2 from Judson's Abstract Algebra textbook. There's a lot of stuff in here reviewing. If you do feel like we're going kind of fast, that is intentional because these ideas of functions and relations and sets are all things we've probably seen in a previous mathematics course prior to Abstract Algebra. We're really just trying to review these important principles as they're relevant for getting ready to study group theory and things like that. Last time we talked about function composition as an operation of functions. In this moment, I want to introduce the idea of the identity function. If we have a function f of x of the form where it maps a to a and it follows the formula f of a equals a, and this happens for all little a inside of capital A, this is referred to as the identity function on, the identity function on a. And typically speaking, we'll actually denote this function as, there's a lot of different ways people would do it. Some people might do like capital ID sub a. You see right here like a little lowercase id sub a. It doesn't matter too much. So this is the identity function for basically two reasons. One, this is the function that just identifies the value plugged in. So whatever value you start with is what you end up with. This is the function that sends a to a, b to b, two to two, five to five, Pikachu to Pikachu, Volkswagen to Volkswagen. It just identifies what it sees. But also in terms of function composition, this is an important observation here to make. If you have a function, let's say f goes from a to b right here, then if you take f composed with the identity function on a, this will just equal f. Which also, this is the same thing as the identity function on b. So if you take f composed with the identity function on a, this will just equal f. Which also, this is the same thing as the identity function on b composed with f. Now the function on the left and the right are not necessarily the same thing because they set a and b is not necessarily the same. But the identity function has the property. Now when you compose it with a function, you'll get back that original function unaltered. So it acts like this identity element with respect to the binary operation of function composition. So with the identity function mentioned, which I should also mention, it's a bijective function. It's one to one and onto. That's very clear. I should also mention that with the identity function, we do stipulate that the set a in conversation is not empty set. It's not the empty set because if it's the empty set, if you take a map from the empty set to the empty set, I mean, although that's possible, you get the empty function, it wouldn't satisfy this relationship. And you have some problems when you start composing these things. So we do require that the domain of the identity function be not empty in this context. Alright, so with that in mind, we're now in a position where we can talk about what it means for a function to be invertible. So an invertible function will say we have f as a function from a to b here. We say that f is invertible or it has an inverse. If there exists a second function, which is called f inverse, it's going to be denoted as f as a superscript negative one. This superscript is not an exponent, like it has nothing to do with multiplication right here. It's just a superscript to represent the inverse function right here. Now, admittedly, we use the negative one here because when one is talking about genuine exponents, x to the negative one means take the reciprocal one over x, which is the multiplicative inverse. In this context, though, f inverse is referring to the composition inverse. This is called the inverse of f and it has the following property that if you compose f with f inverse, so you do this one, then that one, this will give you the identity on a, which is the domain of f. On the other hand, if you compose f inverse, then f, so you do the inverse function first, the composition will equal the identity of b. As we saw previously when we talked about operations of sets and such, the inverse function is going to be the inverse element with respect to function composition. It's the element which when you operate by it with the original element, you'll get the identity. So f composed with its inverse in either order will give you an identity element, an identity function, although which identity you get depends on which direction you go. Now because the domain of f is the codomain of f inverse and because the codomain of f is the domain of f inverse, those two things switch roles there, the output becomes the input and the input becomes the output. Because of that, it doesn't actually matter which order you go. f inverse of f and f of f inverse are both defined, although the order of operations will determine whether you get the identity on the domain or the codomain of f right there. So if an inverse function exists, it'll be unique. This comes from the fact that function composition is associative. And I'm going to delay that argument until we start talking about groups because in a group when you have an associative operation, if inverses exist, they're going to be unique. And so that's going to be the exact same proof one would use in that context. We'll just kind of delay that for right now. So what I want to do is just do a quick example of these invertible functions, functions with an inverse, right? These are some examples we saw in the previous lecture. f of x equals x cubed and g of x equals e to the x. I claim they're invertible if we choose the right domains. So for f of x, we're going to define this from a function from the real numbers to the real numbers. It's defined by the relationship that x maps to x cubed. It has an inverse function, which we're going to call f inverse. That'll go from the real numbers to the real numbers. So what we're going to do is we're going to switch the order here. But as it's the same two sets, you don't maybe notice it. And this will have the relationship that x maps to the cube root of x. In other words, f inverse of x is the cube root of x. Notice what happens when you compose these two functions. If we take f composed with f inverse of x, this would give us the cube root of x cubed. And as these are inverse operations, this gives you an x. Similarly, if you take f inverse of f of x here, this is going to give you the cube root of x cubed, which likewise simplifies to be x here. And notice that this x here is just the identity function. It's the identity function of x. So these functions are in fact inverses of each other. Notice when you compose the two functions, you end up with the identity. So this is the proper inverse function for x cubed. Now if you were to look at it for the exponential function, take g of x to be a map. It's going to go from the real numbers to the positive real numbers. It's given by the formula x will map to e to the x. And so it's candidate for its inverse, g inverse. This is going to be a function which maps from zero to infinity, just the positive reels to all the reels. And what we do is we map from x to the natural log of x. So I claim the natural log of x is the inverse function to the natural exponential. And we can see what happens when we compose these things. If you take g, this is the name of the function this time, g of g inverse of x. Well, we end up with e raised to the natural log of x power. Now be aware that the natural log of x by definition is the, it's the value, which if you raise e to that power, you get its, you get its operand. So the natural log of x is the power, is the number which if raised by e to that power, you're going to get an x. Well, we're now raising e to that power, so this is going to be x. They simplify. Another, another, the other way around g inverse of g here. In this situation, you're going to get the natural log of e to the x. And so with the natural log, you're asking yourself, what power of e gives you e to the x? Well, the x power will do that. So we see that these things are inverses of each other, but do pay attention to the domains, right? The domain of g is the co-domain of g inverse. And the co-domain of g is the domain of g inverse. These things will swap roles. Now in order to make g have an inverse function, we do have to restrict the domain. It's co-domain to be zero to infinity. The problem is if we were to, if we were to take the function g of x equals e to the x, but we allowed its domain to be all, its co-domain to be all real numbers. Then the inverse function, you'd have to switch that to be all real numbers. And unfortunately, the natural log is not defined for all real numbers. What is the natural log of negative one, for example? There's no real number we can assign to that. So do be aware that the existence of inverses depends on the domain and co-domain. And so we have to make sure we're explicit with our functions that we know exactly who's the domain of this function and who's the co-domain of this function as well. Because to kind of further this idea, let me give you another example. Let's take the function h to be from the real numbers to the real numbers. And we take x, we want to map this to x squared. In this situation, we're going to see that there is actually no inverse to this function. It's not invertible in its present format. And that's because there's a problem with the domain and co-domain. For example, because we might be tempted to say something like the following, right? The inverse of h, right? This should be, well, we'll worry about the domain and co-domain in a moment. This should be the square root function, right? Because isn't it true that if you take the square root of x and you square it, that you're going to get back in x? So that feels like that should be the inverse function h composed with h inverse of x. Squeeze in there. That seems right, but you're kind of, if you're not careful, we're ignoring the domains and co-domains. The problem with the natural log of x, not the natural log of the square root of x, is not defined for all real numbers. It's only going to be defined for non-negative real numbers. Again, otherwise we get in problems with imaginary numbers and such. It's only going to be defined from zero to infinity. And therefore, its inverse function, its co-domain, has to be the domain of its inverse. So it would have to be that zero to infinity is the co-domain. And then what numbers come out of this thing, right? Well, you know, we do have some issues. We could try to say like all real numbers and such, but not all real numbers are coming out of the square root. If we're going to be proper, we should mention, we should specify the domain and co-domain of the square root function are non-negatives and non-negatives. And therefore, we should then have h, its domain and co-domain have to agree with the domain and co-domain of the inverse function. I guess I should do that in red. We should switch that to zero and infinity. So by restricting the domain and co-domain, then the square function does have an inverse, in which case that's going to be the square root. That'll be the square root of x, which we see right here. So that way, when you take the square root of x squared, you get back an x. On the other hand, if you take, we did that one, h inverse of h of x, that's going to be the square root of x squared. Now, if one's careless with this, you might say this is x. But actually, if you take the square root of the x squared, that actually gives you the absolute value of x. Notice if you take two or negative two, if you square them, they're both four. And if you take it square, you'll get a positive two back. So the composition here is actually the absolute value of x. Now, wait a second, if I restricted my domains to just be non-negative numbers and my co-domains as well, then absolute value function and the identity function are the same on non-negative integers. And so in that situation, it's like, oh, we did it. If you restrict the domain, then we can get that these things are actually invertible. So the domain and co-domain matters a lot. On the real numbers, x squared is not invertible, but on non-negative real numbers, it's an invertible function and its inverse will be the square root of x.