I ll bet you think numbers are pretty boring, don t you? I ll bet you think numbers have no life. For instance, numbers don t have relationships do they? And if you had no relationships, life would be pretty boring wouldn t it? Well, you re wrong. Numbers do have relationships. Err, yeah right. I was out last Friday night Mr. Thompson I didn t see any numbers out on dates! The dating life of numbers Whether you realize it or not, and evidently you don t, we ve been watching numbers in their relationships all term long! In fact, we ve been playing match-maker since we started the class! We ve been pairing up numbers all along. Can you think of how we ve been doing this? Can you think of any number couples? I ll give you a hint these numbers like to pair up in a very ordered manner ordered pairs? Come on! Think! Where have we been seeing and playing with ordered pairs of numbers? Equations the dating service for numbers So, where do numbers go to find a date? They go to an equation. In an equation, a number can become a pair-the equation is like a match-maker. How s that work? Take this equation y = x 2. Now pretend you are the number 2. What number do you think is your match? Well, plug yourself into the equation and see what number pops out: the number 4 does. So now we know that (at least in this equation) the number 2 and 4 are paired up as (2, 4). What do we call (2, 4)? It is an ordered pair of the x-coordinate and the associated y-coordinate. Relations In math, an ordered pair (x, y) is what we call a relation. Each relation has an input (which we normally think of as x) and an output (y). We have a special name for the input: it is the domain. We call the output the range. I know those are strange names but you need to get used to them. One way I remember which goes with which is alphabet order: x comes before y so domain comes before range. The domain is all the x values and the range is all the y values for a given relation. Identifying domain and range If I gave you a list of ordered pairs, you could tell me what the domain and range of that list is. Let me show you. Here is a list of ordered pairs. Note the curly brackets around the list: it means we re looking at these things as a set: Page 1 of 5
What is the domain? It is the set of x values from the list: {0, 1, 2, 3}. What is the range? It is the set of y values from the list: {-1, 0, 3, 4, 6}. A couple of things to note: 1. Remember, x before y so domain (x) before range (y). 2. If there are duplicates, just list the number once. 3. List the numbers in each set in numerical order. Describing a relation We have several ways of describing a relation. Remember that order is important which numbers go together is what a relation is all about. So we need a way of describing a relation that displays the pairing and ordering. Here are the primary ways we ll be using. Let s use the relation listed above to see each in action. 1. Sets of ordered pairs: (x, y) this is the way I described the relation above: 2. Graphs of ordered pairs on the x-y coordinate plane 3. An equation (you don t see all the numbers but you can find all pairs!) Not all relations have an equation that goes with them this one doesn t. 4. Mapping diagrams: arrows connect the paired numbers; one copy each number. Page 2 of 5
5. Table format: like a T-chart but horizontal not vertical; shows each pair. Input x (domain) 0 1 1 2 3 Output y (range) 3 4 6-1 0 Special relations Not all relations are as useful as others. There is a special category of relations that are so useful, that this entire chapter is devoted to them. They are relations called functions. Now, I m not going to tell you what a function is. I m going to show you four graphed examples of functions and four examples of relations that are not functions. You examine each group and see if you can see the main difference. There is one basic difference between them! YES NO Page 3 of 5
If you are having a hard time seeing the difference, I ll give you a hint: there is something different on the right about the x-coordinates. Each graph on the right as at least two points where Still stuck? Ok, consider the top left graph in the NO set; two of the points are arranged very differently than the similar graph in the YES set. If you see that difference, can you see a similar difference in each of the others? In each of the graphs in the NO set, there are at least two points in line vertically. In other words, there are at least two points (ordered pairs) one directly above the other. This is what makes a relation a function. There are no ordered pairs with the same x-coordinate. Each input has exact one output. For every x there is only one y. When looking at a graph, if it is a function, you can draw a vertical line through the diagram and only hit one point. If the vertical line hits more than one point, it isn t a function. Functions who cares??? Ok so what? Why does that matter at all? Think about it this way: in many situations, if I plug a number into my equation, I want to know, I need to know what the exact one answer is. I don t want to have to decide between two numbers. Predictability Imagine a pop machine is a function. If you put in the correct input (the right amount of money and push the right button) you expect to get a certain output (the correct drink) right? How would you feel about things if when you pushed the Pepsi button you either got a Pepsi or a Dr. Pepper? You really want a Pepsi, but you have a 50/50 chance of getting a Dr. Pepper. No good right? You expect that when you press the Pepsi button you get exactly that. Functions are good because you know exactly what you re going to get out of it; no guess work. They are dependable and predictable. Is our original relation a function? Here is the 1 st relation we talked about. Is it a function? Does every x have only one y? No, it is not a function. Look at the 2 nd and 3 rd ordered pairs. They both have the same x but different y s. One input, two outputs. Not a function. Page 4 of 5
How about this one? Is it a function? Lesson 6-1: Relations and Functions { (-1, 6), (0, 5), (3, 6), (5, 0) } Yes it is! Each x has exactly one y associated with it. Function notation Any equation represents a relation but not all equations represent functions. We have a special way of writing equations that represent functions. The equation y = x 2 + 2 is a function. We say that y is a function of x and write it as y = f(x). The letter f is the name of the function. We could have named it g if we d wanted; then we would have y = g(x). The letter isn t incredibly important although we most often use f(x).and g(x). You will sometimes also see h(x) used. Bottom line is you can use whatever letter you want. Given that y = x 2 + 2 is a function, write it using function notation and the name f: f(x) = x 2 + 2 What is the input of this function? It is x. What is the output of this function? It is x 2 + 2. Evaluating functions We can evaluate a function for a given input value. All that this means is we plug that number into the function and crunch it to see what the answer/output is. Another way of saying this is that f(x) means the value of f at x. Evaluate f(x) = x 2 + 2 at x = -1, x = 0 and x = 1: f(-1) = (-1) 2 + 2 = 1 + 2 = 3 f(0) = (0) 2 + 2 = 0 + 2 = 2 f(1) = (1) 2 + 2 = 1 + 2 = 3 Page 5 of 5