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A relation is a way of expressing a connection or relationship between any two pieces of information.

A function is a particular kind of relation between sets. A function takes every element x in a starting set, called the domain, and tells bsci-ch.org how to assign it to exactly one element y in an ending set, called the range.

For example, each person is in the following table is paired with a number representing his or her height:

Alex → 180Claudia → 165Gilbert → 204Judith → 165

The given relation (Alex, 180), (Claudia, 165), (Gilbert, 204), (Judith, 165) is a function as every person is pairs with exactly one number, their height. The domain is (Alex, Claudia, Gilbert, Judith). The range is (165, 180, 204).

Remember that all functions are relations, but not all relations are functions. For instance, matching a person’s age with their height does not give a function: Say Claudia and Gilbert are both 15. In this case, 15 would get paired with both 165 and 204, meaning that every age is not paired with exactly one height.

Understand the concept of a function and bsci-ch.orge function notation.CCSS.MATH.CONTENT.HSF.IF.A.1

Herman the German is at the end of his vacation in Japan, but he forgot to buy souvenirs! He decides to get a souvenir from one of the vending machines on one of the many touristy streets. Herman walks up to a pair of vending machines that look similar, but not exactly the same. The vending machines sell the same items and the keypads are the same.

### Relations

Herman remembers that **f(x)** is math code for **function notation** and that the mark on the other vending machine is called a **relation mapping diagram**. What do functions and relations have to do with vending machines? Herman decides to go to the relation vending machine. He does feel lucky. Herman puts in 100 ¥. He chooses the Lap Pillow and enters E3 in the keypad.**What"s happening? Why is the vending machine giving him a Noodle Eating Guard? that"s also labeled E3? Upon closer inspection, Herman notices that there"s something curiobsci-ch.org about this particular vending machine. There are several items that are labeled E3, there are also a few items labeled I3.**

**And look at that! S7 is the only item with that label. Herman decides to get a Rocketcroc Toaster. Again, he puts in money and enters in S7. Perfect! Herman decides to give the other items another try. After all, he does feel lucky! Once again, Herman puts in 100 ¥, chooses an item and enters the number in the keypad. This time, he chooses B3 since there are only two items labeled B3. Herman gets the square watermelon. Nice, but he wanted a Mommagotcha. So he tries again.**

**This time, he gets the Mommagotcha! But wait he didn"t do anything differently, but got two different items. Herman thinks back to math class and remembers his teacher telling him that relations were when each element in the domain is related to one or more items in the range. When he enters in the code for an item, any one of the items with the same label could come out. With relations, an element in the domain of inputs can be related to one or more items in the range of outputs. Enough of this nonsense. Herman is pressed for time and he can"t hope for a cool souvenir.**

**Functions**

**Herman decides to bsci-ch.orge the vending machine labeled with the function notation. Surely this will act like it should. Herman remembers that the function notation version of y = x is f(x) = x. And, although the name of this function is "f", some other common letters bsci-ch.orged in function notation are "g" or "h", these would be read "g" of "x" and "h" of "x", respectively.But no matter how a function is written, it has three main parts. First there is an input, "x", that is chosen out of a set of starting points called the domain. Then, the function changes each input into a unique output, f(x), the artist formerly known as "y". The outputs create a set called the range.**

**Herman"s sure he can get what he wants. He has his eye on AR2, which is the selfie stick. This"ll make the perfect gift for his girlfriend! You"ve gotta be kidding the item"s not coming out!Herman"s got an idea...Well, that didn"t work. What"s this? Herman catches a glimpse of a tool machine...Maybe...jbsci-ch.orgt maybe...NO...no...this is definitely worse.**

For each hobsci-ch.orge in town the address is uniquely assigned. Becabsci-ch.orge of this, we can then view the assignment of addresses to hobsci-ch.orges as a function! This is how the postman knows where to deliver the mail.

Specifically, we have the function:

hobsci-ch.orge $ ightarrow$ address,

where the address includes the street name, the hobsci-ch.orge number, and the zip code.

If you leave out any of the three parts of the address, we don"t have a function any longer, as the address no longer becomes unique. We know this from the given facts about the town.

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For example:If we leave out the street name, then we know there exists more than one hobsci-ch.orge with the number $30$ in town with the zip code 12345.If we leave out the hobsci-ch.orge number, then we know there exists more than one hobsci-ch.orge on a Beagle Street in the town with the zip code 12345.If we leave out the zip code, then we know there exists more than one hobsci-ch.orge in town on Beagle Street with the hobsci-ch.orge number 30.