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Relations and Functions - Overview

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x). Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. In many other cases, a function is given by a picture, called the graph of the function. Other functions may be defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.

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So in this section we're gonna be talking about relations and functions. Um, and we're gonna end up talking about this, actually for the rest of this chapter. So this is gonna have a lot of different vocab that you will see throughout eso to get yourself familiar with it now. So the first term is what's a relation? Well, a relation essentially is this any set of ordered pairs? So it could have two ordered pairs, five ordered pairs, 1000 ordered pairs, But that's essentially what it is. So relation is just a set, or I'll put any set of ordered pairs. That's essentially what it is. And a lot of times it might be used to be represent data. So, for example, let's say we had the ordered pairs. Here's my set notation. So let's say we had negative to five. Let's say we had the ordered pair 37 and let's say we have the ordered pair. I don't know. Let's go with 45 So again, there's not a number of specific number ordered pairs that you have to have in here, but essentially just any set of ordered pairs is defined as a relation so ah, function is actually a certain type of relation. It's actually a special case. So ah, function is a relation where each of your values of your or every single X value is Onley paired with one y value. That's the most basic definition, and we'll give in two more terms of it later on. So a function is a relation in which each X value or inch X coordinate is paired. Exactly. One. Why coordinate and the way I kite tend to actually remember. This is essentially my ex values in order. Pairs can't repeat. So here's my little hint. X values cannot repeat in multiple what repairs That's the thank you. Okay, so as long as you can remember, function has all the X value is the only pair with one y value. Then you'll be all set. So based off the relation, we have the negative 2537 and 45 notice. How are ex? Value of negative two is on Lee paired with a Y value of five. So that's good. Are X value of three is only paired with seven. That's good, and our X value of four is only paired with five. So in this case, this relation would be a function. So the key here because some people might say, Oh, whoa, we have two of the same Y values. That's okay as long as your ex values don't repeat and you'll see in some examples that we have moving forward. Um, we'll go over being able to identify what's a function and what's not a function. But essentially, you're looking for Do any of your ex values repeat? Okay, Now one term that's going to be used often when we're talking about functions, especially when you get into higher levels and you start analyzing functions is you're gonna be talking about domain and range. So the domain is the set of your ex values. So it's the set off X values in the relation, and the range is basically just the opposite. This is the set of why values in the relation Well, let's go back to our original example. So what would be our domain? Well, it's essentially is just your ex values. So in this case, our domain would be negative 23 and four. So if we scroll down here for our relation, we had the domain re negative, too. Three and four because those were your ex values. All right, let's go back and take a look. Let's try and determine the range. Well, are why values are 57 and five. So our range in this case is just five and seven. You don't have to repeat any values that, or you don't have to mention any values that repeat. So if we were in the theme our range, we would just call it five and seven. You wouldn't be penalized typically for writing the five twice, But really, you don't have thio. So the key thing here is the domain is the set of X values in the relation on the range is the set of Why values in the relation Okay, the next thing we're gonna talk about is that there's four ways to represent a relation. So here's what those four ways are. Let's get rid of this part here, so the first way is just to write it as we have as a set of ordered pairs. The next way is as a table of values. The third way is as a mapping, which is something that might be new in the last way is by graphic. So these are the four ways that we can represent in the ordered pair are sorry, a relation. So what we're gonna do is we're gonna take our original relation. Negative. 2537 and 45 Let's write it down here just so we don't have to keep scrolling back and forth. So we had negative to five. We had 37 and we had four or five. So what we're gonna do is we're going to represent the same relation using these four ways. So the key thing to know is that each of these four ways air interchangeable, they all represent the same thing. So the first one, a set of ordered pairs. Well, that's basically how we have it written. We're gonna use our set notation, and we're just going to write down our ordered pairs. Negative to five, 37 and 45 So that's how we write it as a set of ordered pairs. Okay, Now, for our table of values. So remember for our table, we're gonna have to columns. We're gonna have our x column and our why column. So let's start with the force first door to pair negative to five. Well, negative two is our X value and five is our white value. And essentially, we'll just keep going through our ordered pairs so far. Ordered pair 37 access three. And why? It's seven. And for the last one, pair 45 x is four. And why is five? So the table values is actually pretty simple and again thes a set of what appears on the table values. They all represent the same thing. Okay, now let's do the mapping. So here's how the mapping looks. You're gonna first start with two. I try and make him ovals as much as possible. They could be circles, but they're anything level on this nature. Okay, We're gonna label the first one our domain, which is X and the second oval will be your range. Which is why So what we're gonna do now in our first oval is we're gonna list all of our X values, and if any of them repeat, we're not going to repeat them. So our X values for each ordered pair, we have negative two. We have three, and we have four. We're even do the same thing for the Y values again, If any y values repeat, we don't need to write it twice. So in this case, we'll just have five and seven again. We won't repeat that last five. So here's how the mapping works. We're going to draw an arrow matching or mapping. We should say the X value to its corresponding. Why value. So, for example, because our first or the pair is negative. 25 We're gonna draw an arrow going from negative to forex toe five for why? So if you're looking at this mapping and you see this arrow, you and no negative 25 is an ordered pair in the relation. Now we're next door. Two pairs, 37 So we're going to draw an arrow from 3 to 7, and our last 400 pair is 45 So we'll draw an arrow from 4 to 5. So if you're looking at this mapping because there's three different arrows, you know there's three different order pairs, okay? And the last wing graphing. So we're just gonna graft you ordered pairs. So the ordered pair negative to five means we go left to and up. Five ordered pair 37 means you go right three and up. Seven. And the order pair 45 means you go right for and up. Five. And so the big thing that we need to know is that these four ways are, like I said, are all interchangeable, but they all represent the same relation. So you need to be able to If you're giving a mapping, be able to graph it, set up a table or set of ordered pairs. Um, if you're giving any of these four ways, you should be able to identify the domain and range from it, and you should also be able to identify whether the relation is a function. Given any notation. Given any of these four ways now, it doesn't mean that you can't if you like. Let's say the set of ordered pairs. If you're given a mapping feel free to write it out as the Senate ordered pairs is that's easier for you. Okay, so the last thing we're gonna talk about is if a relation is a function by using by just looking at a graph, so what we're going to Dio is we're going to use what's called the vertical wind gust and a lot of times well, bravery. VLT. So here's how the vertical line test works. Remember how a relation is a function If none of your ex values, you repeat. So here's how your vertical line test goes. If you congee are a vertical line and it doesn't hit multiple points on the graph, then it is a function. So the vertical line test. If you can draw vertical line and it will never okay two or more points at the same time, then graph represents a function. That's the key. Now some people might like to think of this as the other thing or as the opposite direction if it's not a function. So if you can draw a vertical line and it does hit the graph at two or more points, then the graph does not represent the function. That's how the vertical line test works, So this is kind of a quick test. Just if you're giving the graph. So if I were to set up a quick graph, I won't do anything fancy. So let's say, Here's my Y axis and here's my X axis, and it doesn't even matter what the values of X R. So let's say I had a graph that looked like this. Almost think of it as like an upside down U. So the test of this graph represents a function. Can I draw a vertical line? And it hits the graph at more than one point? And if you know this, each time I graft us for the whole line it on Lee has to graph once. Therefore, this first graph would be, ah, function. So this is a function as opposed Thio. I set up a graph and I forgot to put in my X and Y axis before, so I'll go back and do that just so we're oriented, right? So let's say I had a graph that looked like this. So if I was ajar, Vertical Line noticed how it hits my graph at multiple points and because it hits more than once, that means this graph is not a function. So that's how the vertical line test works. It's one of those where, when you see it visually, it tends to make more sense than the words to it. So again, if you could draw over the whole line and hits the graph and more than one point then, like in this last example, it does not represent a function. So in the next couple of examples, we'll go over being able to identify Domaine Grange wherever function or river. Relation is a function, whether it's from a graph from or repairs from mapping all of the above.