Wize Grade 11 Mathematics Textbook > Introduction to Functions

Functions
What is a function?
A relation is a type of connection among two collections of things.  A function is a relation where every input value xxx is connected to exactly one output value yyy.  

So a function is just a special type of relation.

Example 1
Determine if the following relations are functions or not. 

1.
      

This is a function.  Every value of xxx is connected to exactly one value of yyy. 

2.

This is a function.  3 is connected to only one value, and 5 is is connected to only one value.  It is completely OK that they happen to both be connected to the value of 4. 

3.

This is NOT a function.  4 is connected to both 9 and 5.  

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Representing Functions
There are many ways to describe a connection among things such as using a diagram, table, equation, or graph.  Since a function is a special type of relation they can also be represented using these tools.

Watch Out!
A function could look very different depending on how it's represented.  

In order to call something a function it must satisfy the definition:  
Every value of xxx is connected to only one value of yyy.  
It is OK if multiple values of xxx are connected to the same yyy value.
 
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Vertical Line Test
To determine if a graph represents a function we can check using the vertical line test.

If any vertical line only crosses the graph once, then it is a function.

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Domain and Range of a Function
Domain and Range
The collection of all the possible inputs for a function is called the domain of a function.  
The collection of all the possible outputs of a function is called the range of a function.   

Example 1
Identify the domain and range for each of the functions.

1.
  

Domain:{7,4,1}\{ 7, 4, 1 \}{7,4,1}
Range:{2,3}\{ 2, 3 \}{2,3}

2.

Domain: {−3,−2,−1,1,2,3}\{-3, -2, -1, 1, 2, 3 \}{−3,−2,−1,1,2,3}   
Range: {−1,0,1,2}\{-1, 0, 1, 2 \}{−1,0,1,2}

3.

Domain: {Olivia, Mia, Noah, Lucas, Emma}\{\text{Olivia, Mia, Noah, Lucas, Emma}\}{Olivia, Mia, Noah, Lucas, Emma} 
Range: {15,16,17,18}\{15, 16, 17, 18 \}{15,16,17,18}

Wize Tip
When listing out values for the domain or range it is common to put the list inside of curly brackets

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Example: Functions
Determine if the following relation is a function.  If it is, list the domain and range.

1.
  

Every xxx value is connected to one yyy value so this is a function.  
Domain: {−2,−1,0,1,2,3}\{-2, -1, 0, 1, 2, 3\}{−2,−1,0,1,2,3}
Range: {0,1,2,3}\{0, 1, 2, 3 \}{0,1,2,3}

2.  y=xy=xy=x

Every xxx value is connected to one yy
y value, so this is a function.
Domain:  All real numbers
Range:  All real numbers

3.

This is not a function, 6 is connected to both 7 and 9.

Practice: Functions
Fill in the missing words.
A function is a special type of 
 such that every xxx value is connected to exactly one yyy value.
The collection of all the inputs to a function is called the 
.
The collection of all the outputs of a function is called the 
.
The vertical line test says that any given vertical line should only cross the graph 
. 

I don't know

Practice: Functions
What is the domain of the following function?

Practice: Functions
The following relation is not a function, however if a pair of inputs and outputs were eliminated it could form a new relation that is a function.  Which pair should be eliminated so that it becomes a function? 