What is a function?

When something is "functioning" it means that it is working. When we refer to some machines as "working" it means we expect our inputs match our outputs.  In plain speak, the buttons we press always give up what we expect.

Some real; world examples of machines that should "function" include:
1) A remote control: selecting the right set of keys means that we get the channel we expected
2) A vending machine: selecting the right beverage or snack choice means that the we should get the beverage or snack we expected.
3) A bike gear.  choosing the right gear means the bike should work the way we expected
  
In math, we can express the connection between inputs and outputs as ordered pairs (x,y)

x represents: the inputs, the input values become the domain and these values are considered independent because they are selected by the person inputting the values

y represents the ouputs, these values become the range and they are considered dependent, because they rely on the calculations carried out by the function

Function Notation


Expressing a mathematical statement in function notation sends a powerful message.  It lets anyone using thje statement know that it is a function, therefore, each inut can have one and only one output.  This always reminds me of the Highlander movie because of the tag line: "In the end, there can be only one". 

 

However, for our purposes, in most cases we can simply interchange  with "y".  For example, 

we read f(x) = 2x as y = 2x

Is It a Function?

Testing for a function becomes very simple as long as we keep in mind that each input should always result in the same output.  So we must ask ourselves two questions:

Do the inputs repeat -> (If not, we must assume it is a function)

If the inputs repeat, do the repeated inputs have different ouputs 

-> (If no, then the outputs are always the same for a given input, then it is a function)

IN ALL OTHER CASES, WE MUST ASSUME IT IS NOT A FUNCTION

This test plays out in many ways:

Domain Mapping: (Is this a function?)

Vertical Line Test (Is this a function?)

Ordered Pairs (Is this a function?) (1, 2) (3, 4) (5, 6) (1, 7)

Here are some additional resources about functions.  Be sure to do the practice activity for credit:

Homework Video Tutors:

Identifying functions using a mapping diagram
Identifying functions using the vertical line test
Making a table from a function rule
Finding the range of a function given the domain

Practice Assignment:

Practice Identifying Functions

Answers to activity in this post

(Domain Map: Not a function, the input of zero, leads to two different outputs: -2 and 1

Vertical Line Test: Yes, its a function, evry possible vertical line hits only one value

Ordered pairs: Not a function, the input of 1 is paired to two values: 2 and 7)

Arithmetic Sequences and Series

An arithmetic sequence means that you are always adding or subtracting the same number. Addition and subtraction are two basic arithmetic operations.

The cool thing about knowing that a set of numbers is an arithmetic sequence, is that, once I know the number that I am constantly adding (or subtracting), I can find any term in the sequence.  I can even find the sum of all the terms in my sequence.

Let's look at a sequence we had in class today
(Remember, adding all these numbers gave us the number of diagonals in a 20 sided figure) :

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18

Finding the nth Term

For starters, it looks like we're adding 1 to each previous term. Another way to say this is to say that the common difference (d) is 1.

And this can be broken down as follows:

1st Term = a1 = 2
2nd Term = a1 + d = a2 = 2 + 1 = 3
3rd Term = a2 + d = a3 = 3 + 1 = 4
4rd Term = a3 + d = a4 = 4 + 1 = 5
but rember, a2 can be written as: a1 + d 
so the 3rd Term can actually be written as = a2 + d = a1 + d  + d  =  a1 + 2d   =2 + 1 + 1 = 4
and the 4rd Term can be written as = a3 + d = a1 + d  + d  + d = a1 + 3d =2 + 1 + 1 + 1 = 5
....so, the bottom, line is, you always only need the first term, and the common difference to find any number in a series.  However, take care to notice something else... in the fourth term, we added the difference 3 times, and the third term we added 2 times...it's always one less than the current term we are on

The bottom line
So, to find the nth term, take the first term, and add the difference multiplied by 1 less than the number of terms (another way to look at it is we already have one term, we need to add the same number to get all the rest):
nth term = a1 + (n-1)d

Finding the sum of the first n Terms

The sum of all n terms in an arithmetic sequence is called an arithmetic series.
Let's look at a slightly simpler problem to understand this:

Look at these numbers: 1, 2, 3, 4, 5
Notice that the middle number is 3
Also notice that the average of the first and last terms and well as the two next inner terms, I would also get three:
(1+5)/2 = 3
(2+4)/2 = 3
This is because the numbers are evenly spaced, so the mean and the median (middle number) are always the same.

Notice also, that if I took that mean number and added it 5 times, I get the same answer as if I added all the terms in the original series

3 + 3 + 3 + 3 + 3 = 15 = 1 + 2 + 3 + 4 + 5
OR 3 * 5 = 15

The bottom line
Therefore, if we have the first and last terms, and we find the average, we can just multiply by the number of terms to sum it up.

n * [(a1 + a2)/2]


So...back to the question from today. 
In short, how many diagonals does a 20 sided polygon have?

If we started just knowing the number 20 for the sides and the first term, we could reason that every shape after 3 has diagonals... so, we are looking to find the 17th (ie...20-3) term.
That would be
a1 + (n-1)d
= a1 + ((20-3)-1)d
= 2 + 16 (1)
= 18, the 17th term is 18

And if we wanted to sum up all those terms:

n * [(a1 + a2)/2] ...but first some fancy footwork...

While n = 20, we need to add only 17 terms, so let's represent that  as (n-3).... 20 - 3 = 17

Also, the last term is 2 less than our number of sides, so let's call that (n - 2) .... 20 - 2 = 18 (last term)

And the first term is two, but we can rewite that using the last term 

[n - (n-2)]....20 -(20-2) = 20 -18 =2 (first term)

...So we found the average of  (n-2) and [n - (n-2)]

(n - 2 + n -n + 2)/2 = (n - n + n - 2 + 2)/2 = n /2

So our formula looks like

(n-3)*n/2

= (20-3) * 20/2

= 17 * 10

= 170

Just to check, notice that if we added all the numbers in the series we started with:

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18

...We would get 170.

MORE RESOURCES:

Videos:

Arithmetic Sequences

Arithmetic Series

A slighlty different way of looking at this problem, not using the arithmetic sequence and series concept

CHALLENGE PROBLEM:

Look back at the picture at the top of this post. 

Context: A queen charges each person in her kingdom three gold coins a day.  Every day, some new (unlucky wanderer) enters her kingdom and so her collection of coins is always three more than the day before.

1) How many coins does she collect on the 100th day

2) How many coins, in all, has she collected from day 1 to day 100?

Should everyone have to do math?

It's a fact.  Many people hate math.  And if you ask adults who are not in engineering, medical, business, or similar fields, you're likely to get a comment such as "Math, huh...I never really had much use for it in school". 

Yet, if you really push them, you'll realize lots of people suffer from a case of "I wish" as in "I wish I was better at math" or "I wish I knew how to balance my checkbook" or "If I paid attention in math class, maybe I could have been a....[nsert any job that needs math knowledge here, and to be honest, you'd be hard pressed to find one that doesn't]"

So my questions. 
1) Do you think you need math?
2) If so, what types of math do you think you should study?
3) How do you think the math you are learning will impact your future career goals?

Welcome

I'm so excited to decided to stop by :).  I will be updating this daily with interesting math news articles and cool website links.