## Functions in the Real World

When it comes to recognizing functions in the real world, it is useful to think of the term ''is a function of'' as ''is determined by.'' You see, in a function, no input can be related to two different outputs, otherwise, given the input, we couldn't determine the output. Therefore, the input determines the output, so the output is a function of the input.

For instance, think about those outfits again.

Day |
Outfit |

Sunday |
Plaid Skirt, Brown Blouse |

Monday |
Jeans, Gray Sweater |

Tuesday |
Jean Shorts, Green T-shirt |

Wednesday |
Gray Slacks, Black Blouse |

Thursday |
Flowered Dress |

Friday |
Blue Shorts, White Sweater |

Saturday |
Sweatpants and Sweatshirt |

If it's Tuesday, what outfit will you be wearing? Well, we look to see what outfit to which Tuesday is mapped, and we see that you will be wearing your jean shorts with your green t-shirt. That's not too hard, is it? You can see that your outfit ''is determined by'' the day of the week, so your outfit ''is a function of'' the day of the week.

Pretty neat, huh? You may have been familiar with functions involving numbers, but it's totally cool to see that the function concept can actually be applied to real world situations! Let's consider a couple more examples!

## Examples

Suppose you and a friend are trying to raise money for your favorite charity and decide to have a cupcake sale. You bake a variety of cupcakes and sell them for $3 each. If we let the number of cupcakes you sell be your input and the total amount of money you raise be your output, we can represent this relationship using the following equation:

*A* = 3*x*

We see that the amount of money you raise is determined by the number of cupcakes you sell. This tells us that your total money raised is a function of the number of cupcakes you sell. Once you have a function in equation form, it can be used to determine all sorts of things about the situation. For instance, let's say you and your friend made 150 cupcakes, and you want to know the most money that you can make if you sold them all. To determine this, we simply plug 150 for *x* into the function and simplify.

*A* = 3(150) = 450

We see that if you sell all your cupcakes, you can make $450 for your charity! That would be awesome!

Okay, let's take a look at another example. Let's consider birthdays. After all, who doesn't get excited for their birthday? Suppose you have a group of friends with the following birthdays:

Friend |
Birthday |

Susan |
December 22 |

Andrea |
February 24 |

Beth |
August 19 |

Lisa |
September 9 |

Kathleen |
April 10 |

Observe that one person can't have more than one birthday, and that given one of your friends, you can determine their birthday. Because of this, the birthday is a function of your friend. I bet you never thought of a birthday list as a function! This is extremely useful, because this function allows you to determine your friend's birthday, so you never have to forget their gift!

## Lesson Summary

A **function** is a relationship between two sets of elements in which no element in the first set relates to more than one element in the second set. In mathematics, we represent functions in a variety of ways; we use equations, words, tables, mappings, and graphs. We can also use these representations for functions in the real world. To recognize these functions, it is helpful to think of ''is a function of'' as ''is determined by.'' In doing this, we see that functions show up in the world around us all the time. They are extremely useful in applications for business, finance, engineering, and in everyday events, like remembering someone's birthday! It is definitely worthwhile to be familiar with functions and their uses in the real world.