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Input, Relationship, Output

We will see many ways to think about functions, but there are always three main parts:

Example: "Multiply by 2" is a very simple function.

Here are the three parts:

For an input of 50, what is the output?

Some Examples of Functions

But we are not going to look at specific functions ...... instead we will look at the
general idea of a function.

Names

First, it is useful to give a function a name.

The most common name is "f", but we can have other names like "g" ... or even "marmalade" if we want.

But let's use "f":

We say "f of x equals x squared"

what goes into the function is put inside parentheses () after the name of the function:

And we usually see what a function does with the input:

Example: with f(x) = x²:

In fact we can write f(4) = 16.

The "x" is Just a Place-Holder!

Don't get too concerned about "x", it is just there to show us where the input goes and what happens to it.

It could be anything!

So this function:

Is the same function as:

The variable (x, q, A, etc) is just there so we know where to put the values:

Sometimes There is No Function Name

Sometimes a function has no name, and we see something like:

But there is still:

an input (x)

a relationship (squaring)

and an output (y)

Relating

At the top we said that a function was like a machine. But a function doesn't really have belts or cogs or any moving parts - and it doesn't actually destroy what we put into it!

A function relates an input to an output.

Saying " f(4) = 16" is like saying 4 is somehow related to 16. Or 4 → 16

Example: this tree grows 20 cm every year, so the height of the tree is related to its age using the function h:
h(age) = age × 20
So, if the age is 10 years, the height is:

Here are some example values:

What Types of Things Do Functions Process?

"Numbers" seems an obvious answer, but ...

... which numbers?

... it could also be letters ("A"→"B"), or ID codes ("A6309"→"Pass") or stranger things.

So we need something more powerful, and that is where sets come in:

Here are some examples:

Each individual thing in the set (such as "4" or "hat") is called a member, or element.

So, a function takes elements of a set, and gives back elements of a set.

A Function is Special

But a function has special rules:

It must work for every possible input value

And it has only one relationship for each input value

This can be said in one definition:

Formal Definition of a Function

A function relates each element of a set
with exactly one element of another set
(possibly the same set).

The Two Important Things!

When a relationship does not follow those two rules then it is not a function ... it is still a relationship, just not a function.

Example: The relationship x → x²

Could also be written as a table:

It is a function, because:

Every element in X is related to Y

No element in X has two or more relationships

So it follows the rules.

(Notice how both 4 and -4 relate to 16, which is allowed.)

Example: This relationship is not a function:

It is a relationship, but it is not a function, for these reasons:

Value "3" in X has no relation in Y

Value "4" in X has no relation in Y

Value "5" is related to more than one value in Y

(But the fact that "6" in Y has no relationship does not matter)

Vertical Line Test

On a graph, the idea of single valued means that no vertical line ever crosses more than one value.

If it crosses more than once it is still a valid curve, but is not a function.

Some types of functions have stricter rules, to find out more you can read Injective, Surjective and Bijective

Infinitely Many

My examples have just a few values, but functions usually work on sets with infinitely many elements.

Example: y = x³

The input set "X" is all Real Numbers

The output set "Y" is also all the Real Numbers

We can't show ALL the values, so here are just a few examples:

Domain, Codomain and Range

In our examples above

the set "X" is called the Domain,

the set "Y" is called the Codomain, and

the set of elements that get pointed to in Y (the actual values produced by the function) is called the Range.

We have a special page on Domain, Range and Codomain if you want to know more.

So Many Names!

Functions have been used in mathematics for a very long time, and lots of different names and ways of writing functions have come about.

Here are some common terms you should get familiar with:

Example: z = 2u³:

"u" could be called the "independent variable"

"z" could be called the "dependent variable" (it depends on the value of u)

Example: f(4) = 16:

"4" could be called the "argument"

"16" could be called the "value of the function"

Example: h(year) = 20 × year:

h() is the function

"year" could be called the "argument", or the "variable"

a fixed value like "20" can be called a parameter

We often call a function "f(x)" when in fact the function is really "f"

Ordered Pairs

And here is another way to think about functions:

Write the input and output of a function as an "ordered pair", such as (4,16).

They are called ordered pairs because the input always comes first, and the output second:

(input, output)

So it looks like this:

Example:

(4,16) means that the function takes in "4" and gives out "16"

Set of Ordered Pairs

A function can then be defined as a set of ordered pairs:

Example: {(2,4), (3,5), (7,3)} is a function that says

"2 is related to 4", "3 is related to 5" and "7 is related 3".

Also, notice that:

the domain is {2,3,7} (the input values)

and the range is {4,5,3} (the output values)

But the function has to be single valued, so we also say

"if it contains (a, b) and (a, c), then b must equal c"

Which is just a way of saying that an input of "a" cannot produce two different results.

Example: {( 2, 4), ( 2, 5), (7,3)} is not a function because {2,4} and {2,5} means that 2 could be related to 4 or 5.

In other words it is not a function because it is not single valued

A Benefit of Ordered Pairs

We can graph them...

... because they are also coordinates!

So a set of coordinates is also a function (if they follow the rules above, that is)

A Function Can be in Pieces

We can create functions that behave differently depending on the input value

Example: A function with two pieces:

when x is less than 0, it gives 5,

when x is 0 or more it gives x²

Here are some example values:

x

y

-3

5

-1

5

0

0

2

4

4

16

...

...

Read more at Piecewise Functions.

Explicit vs Implicit

One last topic: the terms "explicit" and "implicit".

Explicit is when the function shows us how to go directly from x to y, such as:
When we know x, we can find y
That is the classic y = f(x) style that we often work with.

Implicit is when it is not given directly such as:
When we know x, how do we find y?
It may be hard (or impossible!) to go directly from x to y.

"Implicit" comes from "implied", in other words shown indirectly.

Graphing

Conclusion

a function relates inputs to outputs

a function takes elements from a set (the domain) and relates them to elements in a set (the codomain).

all the outputs (the actual values related to) are together called the range

a function is a special type of relation where:

every element in the domain is included, and

any input produces only one output (not this or that)

an input and its matching output are together called an ordered pair

so a function can also be seen as a set of ordered pairs

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We will see many ways to think about functions, but there are always three main parts: Here are the three parts: For an input of 50, what is the output? But we are not going to look at specific functions ...... instead we will look at the general idea of a function.

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Biographies

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Math used to be all rote memorization and pencil-to-paper equations disconnected from the real world, but more and more teachers are realizing the importance of making practical, relevant connections in math.

We asked our audience of educators: How do you use the real world to teach math? We've collected some of the most interesting answers, ways teachers are connecting math to the everyday lives of their students.

The Real-World Math Wall

To get her fifth-grade students thinking about the math in their lives-and to head off the inevitable "we'll never use this in the real world" complaint-Samantha Baumgardner, a teacher at Woodrow Petty Elementary School in Taft, Texas, has them bring in a common item and write three ways it relates to math on a notecard. These objects form the class's real-world math wall.

An interesting rule: Once something is placed on the wall, the same item can't be brought in again-pushing students to think outside the box to make real-world math connections. Students bring in objects like playing cards, cake pans, softball score sheets, and cookie recipes. One student brought in a medicine dropper and by way of explanation wrote, "The medicine dropper helps you measure water, and it also helps you with cooking steak."

A School Water Audit

At North Agincourt Jr. Public School in Scarborough, Canada, Krista Dunlop-Sheppard, a bilingual resource and home school teacher for grades 1-6, has her students conduct a water audit at home and at their school. Her students have a wide range of math ability: In a single class, she has students with learning disabilities, students who are gifted, and students who have no diagnoses but need extra support. Modifying a pilot project created by the Toronto Zoo Education Department, students add, subtract, find averages, and measure liquids-like the flow rate of all the water fountains, toilets, and urinals-to measure the amount of water their school uses in a day.

They also interview custodians to discover their daily water usage while mopping floors, and do online research to find out how much water the cafeteria dishwasher uses. When finished, students suggest ways for the school to conserve-like collecting rainwater in a barrel to water plants, and cleaning paint brushes in a bucket instead of using running water. In June, students repeat the water audit and see if the changes they implemented made a difference.

Acting Out Restaurant Scenarios

There's nothing wrong with using money to teach negative and positive numbers, or pizza to introduce fractions, but Justin Ouellette-a third- to fifth-grade International Baccalaureate educator at Suzhou Singapore International School in China-takes these exercises a memorable step further, bringing in menus so students can act out true-to-life restaurant scenarios. Going dutch on dinner and tipping reinforces addition, subtraction, decimals, and percents, Ouellette says.

If you need a good resource: Ouellette has used this free lesson plan about a fictional eatery called the Safari Restaurant.

Integrating Math Into English and History

Making real-world math connections can happen outside of math class, too.

While reading Elie Wiesel's Night, 12th-grade students at Kittatinny Regional High School in Hampton Township, New Jersey, calculate the volume of 11 million pennies to help them imagine the impact of the lives lost during the Holocaust. Ashley Swords, a resource center English teacher for grades nine to 12, uses pennies because they are small and plentiful and allow her to recontextualize a familiar, everyday object.

Students perform other calculations to amplify the impact of the lessons-at the school's football field, for example, they determine the volume of Swords herself and then calculate how many football fields would be needed to bury the 11 million Holocaust victims if they were each Swords's size and were buried in graves 10 feet deep. Guesses ranged from two to 20 football fields, with students finally concluding that it would take about 343 football fields.

Swords knew this lesson was a success when a group of six seniors got emotional after completing it and realizing the magnitude of the deaths in World War II.

Math Recipes

Recipes were perhaps the most popular idea among the elementary teachers who responded to our request for real-world math examples. Fifth-grade teacher Gabi Sanfilippo of Meadow Ridge in Orland Park, Illinois, for example, asked her students to write down one to two ways they used math outside of the classroom during spring break-and more than half of her students wrote that they practiced using measurements and fractions while baking and cooking with their families.

In class, students practice halving, tripling, or quadrupling recipes based on how many people they'd cook or bake for. Most teachers don't actually cook in class, but often students cook at home to practice their new skills.

Another educator, Elizabeth Eagan of Bastrop Independent School District in Texas, brings in a toaster oven to bake in class. She teaches the visually impaired, from newborns to 22-year-olds, using recipes in large print, braille, or audio to show the real-world application of adding and subtracting fractions.

Eagan prints out recipes at 129 percent for her low-vision students or converts the text to braille using braille transcription software, and then prints them using a braille printer. She purchased , a cookbook in both braille and print that makes it easier for families and teachers to help their students if they haven't mastered the tactile language. Students may use electronic magnifying glasses like Pebble and Ruby. For audio, Eagan records herself reading the recipes, has a peer or parent help, or uses the app Seeing AI, which can scan and read recipes aloud.

Grocery Store Field Trip

Many teachers make real-world math connections to grocery shopping. Leanna Agcaoili's second-grade students at Joseph J. Fern Elementary School in Honolulu, Hawaii, are tasked with creating a healthy meal for their family on a $20 budget. On a grocery store field trip, students practice adding and subtracting one- and two-digit whole numbers-and learn about money and budgeting in the process.

Agcaoili says she'll do a practice run in class next year, noting that the first time through some students had difficulty finding their ingredients.

Graphing Favorite Halloween Candy Wrappers

After Halloween, Dottie Wright Berzins, a retired public school teacher, had her students bring in wrappers from their favorite Halloween candy. Depending on the age, the students then created graphs showing their favorite candies.

Younger ages built life-size candy bar graphs, marking the x- and y-axis with masking tape on the floor and using the wrappers to represent the bars. Older students constructed tally charts and paper graphs, and followed ads, tracked which brands advertised more, interviewed peers about their favorite candy, made predictions, and created what-if scenarios-like what if the price of chocolate increased-how would that variable impact the graph?

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Math used to be all rote memorization and pencil-to-paper equations disconnected from the real world, but more and more teachers are realizing the importance of making practical, relevant connections in math. We asked our audience of educators: How do you use the real world to teach math?

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admittedly I'm only basing this on my limited understanding of the hypothetical concept of another spatial dimension, so be gentle if it's inaccurate in some way! just thought I could make the concept slightly easier to interpret. The music used is one of my early remixes of an old favourite, Flaming June by BT.

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What Is The Shape of Space? (ft. PhD Comics)

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Golden ration

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Rafael Araujo's hand-drawn Golden Ratio illustrations are a beautiful fusion of art with science. For the past 40 years, the Venezuelan architect and illustrator has been perfecting his amazing drawings that are all connected by this common theme. Armed with nothing but a pencil, compass, ruler, and protractor he creates drawings that depict the mathematical brilliance of the natural world, and has recently begun to compile renditions of his best work into an adult coloring book that seeks to reconnect humans with nature.

Araujo's illustrations revolve around intelligent patterns of growth that are ruled by the Golden Ratio. This special number, commonly annotated with the Greek letter Phi (?), is equal to 1.618 and can be seen in all sorts of natural spirals, sequences, and proportions. "Phyllotaxis" is the name given for the tendency of organic things to grow in spiral patterns and this number pattern reoccurs so often in nature that some researchers have deemed it a universal law for the perfection of structures, forms, and proportions. From sea shells, leaves, crystals, and even butterfly wings, Phi can be traced throughout our environment, time and time again.

Applying the Golden Ratio to his drawings and leaving the construction lines in the final images allows Araujo to create designs that clearly gravitate around this mathematical framework. Each composition is painstakingly detailed: it can take up to 100 hours for the artist to complete a single one. With a successful funding campaign under way, Araujo's coloring book that celebrates this sacred geometric pattern will hopefully be available to a global audience sometime soon. In the meantime, the campaign is still accepting funding on Kickstarter with plenty of perks available to backers.

Above: Chambered nautilus shell
Chambered nautilus shell DNA Butterflies Double Helix Diaethrea Sequence Butterfly infinite sequence Phoebis, Triple Helix Danaus Plexippus, mirror Blue Morpho, Double Helix Blue Morpho, Sequence The Fibonacci Sequence Fibonacci sequence shell
Rafael Araujo: Website | Facebook | Kickstarter

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Coloring Book Celebrates Mathematical Beauty of Nature with Hand-Drawn Golden Ratio Illustrations

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Rafael Araujo's hand-drawn Golden Ratio illustrations are a beautiful fusion of art with science. For the past 40 years, the Venezuelan architect and illustrator has been perfecting his amazing drawings that are all connected by this common theme.

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Tips for a geometrically superior holiday season.

The following is an excerpt from The Indisputable Existence of Santa Claus by Hannah Fry and Thomas Oléron Evans.
First things first-before your Christmas planning gets under way in earnest, you'll want to get off on the right foot with a geometrically superior Christmas tree.

In some senses, designing the perfect tree is simple. All you need is a neat, symmetrical placement of garland and lights, accented by an equally precise arrangement of spectacular twinkling ornaments.

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But how long does your garland or string of lights need to be to wrap around your tree perfectly? Rather than relying on guesswork, there is a simple equation you can use to help you.

To give you an idea of all the festive glory you can hope for with this calculation, we've included a sketch of your tree on the right.

You did buy a cylindrical tree, right?

Even if you didn't, this version of the formula works for particularly fat Christmas trees that are approximately cylinders, save for the spindly bit at the top which you don't cover in garland anyway.

In red is the garland wrapped around the tree. The height of the cylindrical section is h and the radius is r. If you unravelled the cylinder to look at only the surface of your Christmas tree, it would look something like the following: a rectangle of height h and width equal to the circumference of your tree, 2πr.

The diagonal lines represent loops of garland around your tree. As each line disappears off the edge on the right-hand side, it appears as the beginning of the next diagonal line on the left-like a pine needle PAC-MAN.

Exactly how many loops your tree should have we'll leave to your artistic talents, but once you've made your decision (let's say you have n loops), the length of the garland can be easily calculated by zooming in on one of these diagonal lines.

Since you want everything to be evenly spaced, the vertical distance each loop must cover is h/n. The width is just the circumference of your tree, 2πr, and, since the length l of the garland on this particular loop forms the hypotenuse of a right-angled triangle, Pythagoras does the rest for us:

Who said Pythagoras would never be useful?

Taking all n loops into account, the total amount of garland required, L, is given by the following formula.

"But hang on!" we hear you cry. "My tree isn't cylindrical!" If anything, it looks more like a cone. Typical mathematicians making sweeping unjustifiable assumptions...

Well, hold on to your accusations for now. While you can use the cylindrical version as an upper limit on your required amount of garland, even if your tree gets a lot skinnier at the top, we've also done the calculation for a tree that takes the more traditional shape of a cone.

The math isn't for the faint-hearted, but if you're keen not to waste garland you can find the equation in the book.

The Indisputable Existence of Santa Claus: The Mathematics of Christmas

For the rest of us, happy with our geometrically dubious trees, we can move on to the matter of ornaments. Spherical ornaments seem to be the traditional decorations with which to adorn a tree and in any other setting we're big fans of spheres. But while they may catch the light rather beautifully, to a mathematician they're just not very...festive.

If we're being honest with ourselves, we can all agree that there is nothing more Christmassy than corners.

Not just any corners, of course. We want neat, symmetrical, mathematically superior corners. Which is why, for our tree decorations, we're turning to our five favorite shapes; objects that have been admired and studied by geometers for thousands of years.

Ladies and gentlemen, may I introduce you to...The platonic solids:

The platonic solids are 3D shapes where every face is a regular polygon. Around a face, every side is the same length and every angle is the same size. Looking at each shape as a whole, every face is identical and every corner is the same as every other.

Our platonic solids are: (top row, left to right) the tetrahedron, the hexahedron (which often goes by its street name of "cube"), the octahedron, (second row) the dodecahedron, and the icosahedron.

These five beauties are the only three-dimensional shapes that fit our criteria. We know this, not because we've checked all possible three-dimensional shapes, but because we can prove it. To find out how, you'll have to read the book.

Any of these five platonic solids would look marvellous on a mathematical Christmas tree. After careful consideration, we reckon the hexahedron offers the perfect aesthetic balance between simplicity and cornery goodness.

You can make your cubic ornaments using the template to the right.

With your tree bedecked with fantastically festive Christmas cubes, you're sure to be the talk of the town.

Hmm...For some reason, our editor seems to think that cubical ornaments are insufficiently ambitious. Well then, in case you also can't appreciate the supreme elegance and mathematical purity of the cube, we've put together some instructions for making your very own stellated icosahedral ornaments.

The icosahedron is the basis for this shape. Technically, a stellation is where you extend out the planes of each face until they meet beyond the boundary of the original shape. In this case, it's just like adding a point to each of the faces of your icosahedron.

The end result here is a 20-pointed three-dimensional star that will look superb as a sparkling ornament or, when supersized, perfect taking the place of pride at the top of your tree.

Here's how to make it.

This origami-style technique involves making a series of modules from smaller pieces of paper and then slotting them together to create the final shape.

You need to start off with a square piece of paper. Some people suggest using Post-it notes, others recommend tinfoil, but for that extra Christmassy glitz, we think metallic wrapping paper is the way to go.

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Carry on slotting modules together in the same way, securing with glue if necessary, until all 30 are in place. The final few will be a little tricky, but the shape of the icosahedron should emerge naturally from the creases in the paper.

And there you have it. Simply repeat this entire process 20 or so times to create 600 modules which will slot together for a complete set of ornaments, plus 30 more with larger squares of paper for the star on the top of your tree. And, assuming it's still December, you'll have a festive centerpiece bedecked in mathematically glorious decorations that will be the envy of all of your friends at your Christmas party.

At which point you can just sit back, add some ice regular hexahedrons to your rum punch, and marvel at your creation.

Adapted from The Indisputable Existence of Santa Claus by Hannah Fry and Thomas Oléron Evans. Copyright © 2016, 2017 by Dr. Hannah Fry and Dr. Thomas Oléron Evans. Published in 2017 by The Overlook Press, Peter Mayer Publishers Inc. www.overlookpress.com. All rights reserved.

Meet the Writers

@fryrsquared

Hannah Fry, Ph.D., is a lecturer in the Mathematics of Cities at the Centre for Advanced Spatial Analysis at UCL, and is co-author of The Indisputable Existence of Santa Claus (The Overlook Press).

@Mathistopheles

Thomas Oléron Evans, Ph.D., is a Teaching Fellow for the Smart Cities and Spatial Data Courses at the University College London, and the co-author of The Indisputable Existence of Santa Claus (The Overlook Press).

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