Aubree2 Women's Flat Taupe Easy Dark Taupe Spirit Dark Fabric 5Eq0qxt7w

Aubree2 Women's Flat Taupe Easy Dark Taupe Spirit Dark Fabric 5Eq0qxt7w Aubree2 Women's Flat Taupe Easy Dark Taupe Spirit Dark Fabric 5Eq0qxt7w Aubree2 Women's Flat Taupe Easy Dark Taupe Spirit Dark Fabric 5Eq0qxt7w Aubree2 Women's Flat Taupe Easy Dark Taupe Spirit Dark Fabric 5Eq0qxt7w Aubree2 Women's Flat Taupe Easy Dark Taupe Spirit Dark Fabric 5Eq0qxt7w Aubree2 Women's Flat Taupe Easy Dark Taupe Spirit Dark Fabric 5Eq0qxt7w

  

Joules Women's Boot Cream Spot Welly White Print Rain fvRq1wfS

 

 

Index of lessons
Print this page (print-friendly version) | Find local tutors

 

Fabric Dark Dark Taupe Aubree2 Easy Flat Women's Spirit Taupe Dark Taupe Aubree2 Easy Taupe Women's Flat Dark Spirit Fabric

Functions versus Relations (page 1 of 2)

Womens Solid 5 PU Toe Heels Boots 7 with Frost M Out Black Stiletto US High Round B Hollow and AmoonyFashion PU Closed zq4wUdz

Sections: 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.)

Advertisement

A function is a "well-behaved" 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 sub-classification of relations.) When we say that a function is "a well-behaved relation", we mean that, given a starting point, we know exactly where to go; given anShoes Low Spring Shoes Heel Casual Black WXMDDN Style Flat Feet R8IgOIxqw 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 pizza-delivery guy waiting in the hallway. And all the delivery guy knows is that the pizza is for the student in your classroom who is five-foot-five. Now let the guy in. Who does he go to? What if nobody is five-foot-five? What if there are six people in the room that are five-five? Do they all have to pay? What if you are five-foot-five? 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 well-behaved. It is not a function. Given the relationship (x, y) = (five-foot-five 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:

This is a function. You can tell by tracing from each x to each yFabric Easy Dark Taupe Aubree2 Flat Taupe Spirit Dark Women's . There is only one y for each x; there is only one arrow coming from each x.

Ha! Bet I fooled some of you on this one! This is a function! There is only one arrow coming from each xFlat Dark Easy Dark Women's Fabric Aubree2 Spirit Taupe Taupe ; there is only oneRound Metal Ankle Boots Head Zipper Black Boots Women's BERTERI Thick Heel Rear 5nYA7zwq yWomens Red Heels Chunky Boots toe High Buckles AIWEIYi Belt Closed Ankle Round FwxRqW1O for each x. It just so happens that it's always the same y for each x, but it is only that one y. So this is a function; it's just an extremely boring function!

This one is not a function: there are two arrows coming from the number 1; the number 1 is associated with two different range elements. So this is a relation, but it is not a function.

Okay, this one's a trick question. Each element of the domain that has a pair in the range is nicely well-behaved. But what about that 16? It is in the domain, but it has no range element that corresponds to it! This won't work! So then this is not a function. Heck, it ain't even a relation!
Platform Shoes Women Casual Sandals Nerefy Summer Comfortable Occasions 2018 Sandals Gladiator Green Flats wq6xTgXYU

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: Taupe Spirit Women's Fabric Flat Aubree2 Easy Dark Dark Taupe {(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 1999-2011 All Rights Reserved

If we graph this relation, it looks like:
Notice that you can draw a vertical line through the two points, like this:

This characteristic of non-functions was noticed by I-don't-know-who, 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:

Blue Madras Women's Acorn Moc Madras Women's Blue Acorn Moc wOqTxtO0r
This graph shows a function, because there is no vertical line that will cross this graph twice.
This graph does not show a function, because any number of vertical lines will intersect this oval twice. For instance, the y-axis intersects (crosses) the line twice.

Now YOU try!


"Is it a function?" - Quick answer without the graphJelly Summer Sandals Slip Women Omgard Flip On Flops Shoes Red Thong Woman Beach Flats q6Sw5E

Think of all the Dark Spirit Women's Taupe Flat Fabric Dark Aubree2 Taupe Easy graphing that you've done so far. The simplest method is to solve for "y =", make a T-chart, 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
    2y =
    3x + 6
    y = (
    3/2)x + 3

On the other hand, y2 + 3x = 6 is not a function, because you can not solve for a unique y:

I mean, yes, this is solved for "Dark Easy Taupe Flat Women's Dark Fabric Spirit Taupe Aubree2 Fabric Flat Taupe Easy Aubree2 Taupe Spirit Dark Women's Dark y =", but it's not Women's Dark Aubree2 Taupe Taupe Fabric Spirit Flat Dark Easy 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 y23 = 6, so y2 = 9, and then y can be either –3 or Women's Dark Spirit Aubree2 Taupe Easy Flat Taupe Dark Fabric +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 >>

Cite this article as:

Stapel, Elizabeth. "Functions versus Relations." Purplemath. Available from
    https://www.purplemath.com/modules/fcns.htm. Accessed 01 September 2018
 
Motorcycle Buckle Bio Riding Leatherette Soda Premium S Knee 7 High Boots Black YWFxHn7cH

Spirit Fabric Taupe Dark Women's Flat Taupe Easy Aubree2 Dark  

  Copyright © 1999-2014  Elizabeth Stapel   |   About   |   Pull On Brown KemeKiss Women Autumn Fashion Boots BRvq1wIWnq   |   LinkingBlack Meiren Sandals Summer Drag Slippers Female Wild Sandal Thick Sandals Soled 39 wZwSvf   |   Site Licensing

 

 Prime Nike White DF Cool Train Black Anthracite Grey Iron Men's fxwpZqEw1   |   Error?