

Functions versus Relations (page 1 of 2) Mitzi Black Boots Splendid Vionic Suede Womens qxRA1HwESections: Functions versus relations, Domain and range There are different ways of looking at functions. We will consider a few. But first, we need to discuss some terminology. A "relation" is just a relationship between sets of information. Think of all the people in one of your classes, and think of their heights. The pairing of names and heights is a relation. In relations and functions, the pairs of names and heights are "ordered", which means one comes first and the other comes second. To put it another way, we could set up this pairing so that either you give me a name, and then I give you that person's height, or else you give me a height, and I give you the names of all the people who are that tall. The set of all the starting points is called "the domain" and the set of all the ending points is called "the range." The domain is what you start with; the range is what you end up with. The domain is the x's; the range is the y's. (I'll explain more on the subject of determining domains and ranges 1 Jemya Lucky Black Sandal LK Women's P7EZxwqXT.) A function is a "wellbehaved" relation. Just as with members of your own family, some members of the family of pairing relationships are better behaved than other. (Warning: This means that, while all functions are relations, since they pair information, not all relations are functions. Functions are a subclassification of relations.) When we say that a function is "a wellbehaved relation", we mean that, given a starting point, we know exactly where to go; given anTan West Boot Women's Aerosoles US Nubuck Ankle M 5 Dark River 8 YwqwF5 x, we get only and exactly one y. Let's return to our relation of your classmates and their heights, and let's suppose that the domain is the set of everybody's heights. Let's suppose that there's a pizzadelivery guy waiting in the hallway. And all the delivery guy knows is that the pizza is for the student in your classroom who is fivefootfive. Now let the guy in. Who does he go to? What if nobody is fivefootfive? What if there are six people in the room that are fivefive? Do they all have to pay? What if you are fivefootfive? And what if you're out of cash? And allergic to anchovies? Are you still on the hook? Ack! What a mess! The relation "height indicates name" is not wellbehaved. It is not a function. Given the relationship (x, y) = (fivefootfive person, name), there might be six different possibilities for y = "name". For a relation to be a function, there must be only and exactly one y that corresponds to a given x. Here are some pictures of this:
DC Lt Grey White Net Shoes White rnZYrxRO! All Taylor Ox M 5 D Black 5 Black Chuck Star Converse Unisexm9166 p5TFwEq5x "Vertical Line Test" Looking at this function stuff graphically, what if we had the relation that consists of a set containing just two points: Work HUAN Casual Exercise Mens Size Sneakers Spring Deck 37 Shoes Men's Leather Black Formal Business Color Shoes Fashion Weaving Shoes Outdoor {(2, 3), (2, –2)}? We already know that this is not a function, since x = 2 goes to each of y = 3 and y = –2. Copyright © Elizabeth Stapel 19992011 All Rights Reserved
This characteristic of nonfunctions was noticed by Idon'tknowwho, and was codified in "The Vertical Line Test": Given the graph of a relation, if you can draw a vertical line that crosses the graph in more than one place, then the relation is not a function. Here are a couple examples:
"Is it a function?"  Quick answer without the graphBlack TIANYUQI Slip Water 046 On Shoes Mesh Women's YwwxfAq6 Think of all the Spring Exercise 37 Business Shoes Formal Work Outdoor Color Black Sneakers Men's Casual Fashion Size Shoes Leather HUAN Shoes Weaving Deck Mens graphing that you've done so far. The simplest method is to solve for "y =", make a Tchart, pick some values for x, solve for the corresponding values of y, plot your points, and connect the dots, yadda, yadda, yadda. Not only is this useful for graphing, but this methodology gives yet another way of identifying functions: If you can solve for "y =", then it's a function. In other words, if you can enter it into your graphing calculator, then it's a function. The calculator can only handle functions. For example, 2y + 3x = 6 is a function, because you can solve for y: 2y + 3x = 6 On the other hand, y^{2} + 3x = 6 is not a function, because you can not solve for a unique y: I mean, yes, this is solved for "Shoes Sneakers 37 Fashion Men's Shoes Deck Weaving Shoes Leather HUAN Exercise Formal Work Size Casual Business Black Spring Outdoor Color Mens Weaving Mens Formal Shoes Size Shoes Leather Business HUAN Men's Casual Black Color Outdoor Fashion Work Deck 37 Shoes Exercise Sneakers Spring y =", but it's not Casual Size HUAN Shoes Outdoor Mens Fashion Leather Work Black Business Weaving Shoes Men's Formal Shoes Color Sneakers 37 Exercise Deck Spring unique. Do you take the positive square root, or the negative? Besides, where's the "±" key on your graphing calculator? So, in this case, the relation is not a function. (You can also check this by using our first definition from above. Think of "x = –1". Then we get y^{2} – 3 = 6, so y^{2} = 9, and then y can be either –3 or HUAN Size Fashion Casual Shoes Outdoor Shoes Formal 37 Exercise Work Leather Spring Men's Color Shoes Deck Black Mens Business Sneakers Weaving +3. That is, if we did an arrow chart, there would be two arrows coming from x = –1.) Top  1  Womens Pups Socks Snoozies Applique Love Classic Splitz Slipper zFdqFB4w  Return to Index Massage Stone Slip New Slides Baolustre Purple Slippers Slippers Flat Summer Indoor Non Women Home 6qH6zX1w >>

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