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If you repeat an x-value it has to have the same y-value​
A function may not have two y-values assigned to the same x-value, such as (2,5), (2,7).

A function may, however, have two x-values assigned to the same y-value, such as (2,5), (3,5)

Examples of Functions  1.  (2,5) (3,6) (4,7) (8,9)  

Examples of Functions 2. (2,5) (3,5) (4,7) (9,10)

Non-examples (2,5) (3,6) (4,7) (4,9)  
Notice 4 has two different y values 7 and 9

Today we are going to look at “What is a function? “by looking at ordered pairs. Do these ordered pairs create a function? The answer is yes but how do you know? Let’s look at the rules and then we will come back and explain this example a little bit more. What are the rules to decide if you have a function or not? First, each X or input has only one unique Y output. That is hard to think about so I have an analogy to show you. Think about a coke machine. The X values are like the buttons on the front of the machine if you press the coke button you will get a coke out. If you press the diet coke button you will get a diet coke out and if you press a sprite you will get a sprite out. Can you press a coke and get a diet coke out? No that means it is not functioning correctly therefore it is not a function. A coke button cannot produce a different type of output other than a coke. Now let’s go back and look at with ordered pairs. We have an X value of 1 and we get out a 2 so that is a unique output We push the 2 we get a 5, push the 3 get a 10, we push an 8 get out a 2 that’s OK because we can have different coke buttons getting the same output. We have a 6 getting out a 5 Notice I have two different buttons producing a 6 but what we can’t have is a function as it stands what happens if I say we are going to have the two button and we will get a 10? Now notice I have 2 number 2s (Two different buttons that are coke buttons) First time I get out a 5 the second time I get a 10 that’s not possible I can’t push the same button and two get different drinks. That’s not a function. You look for repeating X values. If you repeat an X value you have to repeat the same Y value. That’s how you decide if you have a function or not given ordered pairs. Thanks for watching 

What is a function?

Remember with ordered pairs the x is writen first (x,y)
What is a function?
A function is a set of ordered pairs in which each x-element has a unique y-element associated with it.

This is what is meant by, "each x-value has only one unique y-value." 
For example: a Coke Machine

If you press the Coke button you get a Coke
Press the Sprite you get a Sprite
Press Diet Coke and get a Diet Coke
If you press the Coke and get a Sprite then it is not a unique X value

Each X has one unique Y value 
A function may be written in function notation as follows:
f(x) = 
Let’s look at an example
f(x) = 2x
This simply means that the function, f, with the input of x, produces an output of 2x.  Replace all the x values with the assigned input.

Example:  if x = 3 then we write f(3) = 2 (3) = 6,
so f(3) = 6

The f(x) is the dependent variable. It  is the functions output known as the range. It determines the resulting values for the output variable.

In this example the x is the independent variable. It is known as the domain, because it is your allowable values for the independent variable.

Transcript |What is a function?|

vending machine to help explain a function in math
There are several methods for writing functions. 

Function written using a T -table

t table function 1
vertical line
This is a function because each x has a unique y value. Notice 1 and 8 have the same y value. This is still a function. 
t table
function t table
This is not a function, because the x value 2 does not have a unique y value. 2 has a value of 1 and 3.  Since it is not a function, we call it a relation.

Function Symbolically
Both of these sets are functions because each item in the first set is paired with exactly one item in the second set.
​This pair of sets is not a function because a does not have a unique value. In this example, a can have a value of 3 or 4.  This is called a relation.
Function as a graph

Functions are very easy to graph. You simple plug in x values to calculate the corresponding y value. 
For example:  f(x) = 2x + 5  

Graph the following x values: 2,4,6,

Plug in the x value 2*2 +5 = 9
4*2 +5 = 13
6*2 +5 = 17

You then just plot the numbers. You then can use the vertical line test to see if the graph is a function. If the vertical line ( or pencil) touches the line more than once it is not a function.

Pencil only touches once so it is a function.
Pencil  touches more than once so it is not a function, but it is a relation.