 In this video, we're gonna talk about properties of equality, how they relate to functions and how that relates to solving single variable equations. So we talk a lot about functions in math 1050. And so let's suppose we have a function, let F be a function, all right? And let's also say that some numbers A comma B are inside of the domain of F such that we have that A equals B. So we have two different numbers inside the, I shouldn't say two different numbers, we have the same number just represented twice. Well, if we have a function, then what's gonna happen is when you take F of A, and you compare this F of B, if A equals B in the first place, then F of A is gonna equal F of B. That is when we evaluate the function on the same value, it's gonna get the same output, the same input produces the same output over and over and over again. This is what it's all about to be a function. We have learned with the vertical line test that given a unique, give a specific X coordinate, there's always a unique Y coordinate attached to it that the same X coordinate cannot produce multiple Y coordinates. That's what it means to be a function. So let me try to clarify, when might this property fail? So consider the following sort of non-example here. Let our function F, little take as inputs fractions. So let's say we have a fraction of the form P over Q and our function F is gonna have the following rule that it reports back just the numerator of the fraction. F of P over Q equals P. That seems like a perfectly good function, right? But what about the following? Note that if I were to take, for example, the fraction one half as fractions, this is the same thing as two forces, isn't it? One half versus two forces, it's the same thing, but if you take this function of the valuation here, this so-called function, right? F of one half, well, if you return just the numerator, you get a one. That's not the same thing as two, which is the numerator of two fours. And you can see what happens here is that this assignment, it depends on the representation of the number, right? Do we want a function, which if you give it three, it says seven, but on the other hand, when you give it two plus one, it says five, right? Do we want a function that does something like that? Well, guess what? Those type of things are not functions. This is not a function because functions have this property that if you take A equals B, and it doesn't matter if A and B look the same, if they are actually equal as numbers, then it must be that the output of the function will be the same. And so this right here is what we describe as being a well-defined function. So the function is well-defined. Is it some terminology we use to describe this type of thing? That doesn't matter how you look at it, the number, the output will be independent of that. And so this is true for any function under the sun, there's no function that doesn't have this property because essentially this property is what makes it a function. The same input, even represented differently, will have the same output each and every time. So when you apply this to the functions of arithmetic, right? So we could talk about there's a function that'll send X to X plus C. So there's an addition by C function. There's a function that'll send X to X minus C, the subtraction function. There's the function that'll send X to C times X are the function that'll send X to X divided by C. We can make functions out of the four arithmetic operations. And so if we apply this principle to these functions, you get the following that if A equals B to begin with, then A plus C will equal B plus C for any number C. And that's because we're just applying the addition operation there. Same thing for subtraction. If you subtract to both sides, you get the same quantity again. If you were to times both sides by some constant C, equality will be preserved. If you divide by C, our equality will be preserved because these are special types of functions. It's nothing particularly special about the arithmetic operations. It's just if you apply the function, whether it's addition, subtraction, multiplication, division or something else, equality will be preserved. Now there's a few things I should say about this that when a function is one to one, this means that actually the solution of the equation will be preserved, but it is possible for the equation to enlarge itself, right? So if you take an equation like X equals two, then clearly the only solution to this equation is X equals two. Now if you couple that with the function F of X equals X squared, and then you apply that, you're gonna get F of X equals F of two, which is to say that X squared equals four. In the process of solving this one, it turns out there's actually two solutions. You get X equals plus two and minus two. So you'll notice that the original equation was X equals two. When you apply a function, in this case, the squared function, you can solve it, you can solve the new equation, and you do retain the original solution of two, but sometimes you gain extra solutions. Ones that are solutions of a subsequent equation, but not the original equation. I often like to refer to this as party crashers, someone who wasn't invited to the equation. One has to typically look out for something like this as you're solving equations, but this is only an issue when you work with a non-invertible function. That is a function that's not one to one. X squared has the property that two squared is equal to negative two squared, and this is how negative two kinda snuck in there. It's always a good practice to check your solution at the end of solving an equation. To double check that something like this doesn't happen. One is just good to check your algebra in the first place because we all make mistakes, but oftentimes we have to check our answer to see if it worked, because even if we did everything right, we might have discovered a number that wants to be a solution, but isn't really because of some non-one-to-one function that's used. Some other things I should mention here is that once in a while when you're solving an equation, you end up with something called a contradiction. Let me put that in one line there. You get a contradiction. What do I mean by a contradiction? A contradiction would be an equation of the following form. You end up with something like two equals negative five. You get an equation which is absolutely false. There's no assignment of the variable X that could make this thing true, and that's typically because you get two numbers which are not equal to each other. This happens when you try to solve equations on, this happens when you try to solve an equation sometimes. Like if you had some equation like two X plus one equals three, using the properties of addition that we learned above, we could subtract the number one from both sides, thus combining things, plus one, minus one, those cancel out, three minus two ends up with a, sorry, three minus one gives us a two, two X equals two. You can then use the multiplicative property here, divide both sides by two. We're dividing by two to cancel out the multiplication by two on the left-hand side. We're applying the inverse operation, and we end up with X equals one, right? The solution here. You can check this solution easy enough. Notice two times one plus one equals two plus one, which is equal to three. We checked the answer. No party crashers here. You're never gonna get a party crash when you use a one-to-one function like addition, subtraction, multiplication, division. That does happen when you do other non-one-to-one functions like squaring, sine, cosine, something like that. So getting back to the idea of a contradiction, right? A contradiction occurs when you get something that's absolutely false. No assignment of the variables can make this true, such as two equals negative, two equals negative five here. And in this situation, it turns out there is no solution. The solution would be nothing, right? If ever you're trying to solve an equation, you end up with a contradiction. It means there is no solution. So for example, if you had something like x plus two equals negative x plus x, you might do something like the following. I'm gonna subtract x from both sides. What's good for the goose is good for the gander. You cancel the x on the left. You cancel the x on the right. And then you're left with exactly this equation right here. This would then tell you you have no solution. On the other hand, sometimes we end up with things which we're gonna call identities. What is an identity? An identity is going to be an equation that's always true, irrelevant of the function assignment. So as an example of such a thing, what if we ended up with something like three equals three? We go about solving our equation. We end up with an identity. Then it really tells us that, okay, there's no restriction on our variable here. Like so for example, if we had x plus three equals x plus three, and then you subtract x from both sides, again, just using an additive property of addition, the x's cancel out, we end up with three plus three. In this situation, there's no restriction on the variables. And we see that all numbers, all numbers solve this equation. All real numbers can work here. There's nothing that prevents it from happening. So contradictions and identities don't seem to happen that often. But when you're solving equations, that can be a possibility here. Most likely though, you're gonna see things like the following. You have an equation. And as long as you do the same function to both sides of the equation, you'll retain equality. And so what we do is then we just apply inverse operations, one by one, getting rid of operations attached to our variable x until we get our solution. But like I said, also watch out for that. Even if we do everything correctly, party crashes can't enter the mix. This will be because our functions are not one-to-one. So it's always good practice to check your answer when you're done. When it comes to working with inequalities, you have a lot of the same principles, right? Because of the following. Again, let F be a function. And in this situation, we're gonna assume that A and B are numbers inside the domain of F such that little A is less than B here. Now in this situation, we have to break things up a little bit because it's applying the function F to both sides, won't necessarily give us the inequality we want. Instead, we have to kind of break it up into two cases. If F is an increasing function, whoops, if F is increasing, in that situation, we're gonna get that F of A is less than F of B. That's what we get right there. On the other hand, if F is decreasing, if F is decreasing, then we're gonna get that F of A is actually greater than F of B. And if you look at the definition of increase and decrease, and then it becomes apparent why this is, a function is increasing on an interval A to B by definition if, as the X coordinates get bigger than the Y coordinates get bigger. On the other hand, a function's decreasing on the interval A to B if when the X coordinates get bigger, the Y coordinates get smaller. So this is just applying the definition of increase and decreasing, but it has an important application towards solving inequalities. If you have an inequality, A is less than B, and of course there's appropriate modifications for less than or equal to, greater than or greater than or equal to. If the function's an increasing function, you apply it to both sides, the inequality will be going the same direction it was before. But if you have a decreasing function, then the inequality will be reversed. And so I'll kind of mention to you that these four arithmetic operations, addition, subtraction, multiplication, division, these are all special cases of linear functions, functions of the form f of X equals m X plus B. So all of these arithmetic operations are just special cases of the linear functions because if you have a plus or minus, that can, that just means you're adding or subtract, if you have a plus or minus, if you have an addition, subtraction, that means you're either adding a positive or adding a negative, right? And if you're dividing or multiplying, it just depends on the number bigger than one because if you divide, this is multiplying by the reciprocal, this encodes all the possibilities here. Now, if you're adding or subtracting these principles right here, that just means that your m is the number one. And so you have a, your slope is positive one. And if a line has positive slope, it's an increasing function. So if you add or subtract c, it'll be increasing function as a linear function. Therefore, the inequality is preserved. On the other hand, if you are multiplying or dividing by c, you're basically assuming that the b value is zero and then m is either greater than one or negative, or, you know, it's greater than one or less than one or something like that. And so then what you see happening here is, is, is this function mx, is it an increasing function or decreasing function? It depends entirely on the number, the slope there, which here it's listed as an m, up here it's listed as a c, right? If c is a positive number, then the linear function will be an increasing line and therefore equal, inequality is preserved. But on the other hand, right, if c is a negative quantity, then the inequality gets reversed when you multiply it by it, because in that case, you've now times by a negative slope, which is actually a decreasing function. So, you know, we often learn that, oh, solving inequality is just like solving the equation. It's just when you multiply by a negative, it switches directions. That's, that is true, but that'll be true in general for decreasing functions. When the function's decreasing, well, you have to switch the direction of the inequalities. And so this is great. So for example, when you use things like f of x equals the square root of x, the square root function is an increasing function. So this property would apply. If you take something like g of x equals e to the x, the natural exponential, that's an increasing function. Therefore, this rule applies when you solve inequalities. On the other hand, if you take something like h of x equals x squared, this function's neither increasing nor decreasing. It's increasing from zero to infinity, but it's decreasing from negative infinity to zero. And since the function switches its monotonicity, that makes solving inequalities involved a little bit more complicated, which is why we'll talk about quadratic inequalities a little bit more in the future. But other than that, solving an inequality works just the same way as one would solve a equation. Just make sure you pay attention to if the function increasing or decreasing.