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)