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Wize High School Algebra I Textbook (Common Core) > Relations & Functions

Functions

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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 x\bcfi{x} is connected to exactly one output value y\bcfi{y}. 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 xx is connected to exactly one value of yy.

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 xx is connected to only one value of yy.
It is OK if multiple values of xx are connected to the same yy 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 \}
Range:{2,3}\{ 2, 3 \}


2.

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

3.
Domain: {Olivia, Mia, Noah, Lucas, Emma}\{\text{Olivia, Mia, Noah, Lucas, Emma}\}
Range: {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 xx value is connected to one yy value so this is a function. Domain: {−2,−1,0,1,2,3}\{-2, -1, 0, 1, 2, 3\}
Range: {0,1,2,3}\{0, 1, 2, 3 \}

2. y=xy=x
Every xx value is connected to one yy 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.
  1. A function is a special type of
    such that every xx value is connected to exactly one yy value.
  2. The collection of all the inputs to a function is called the
    .
  3. The collection of all the outputs of a function is called the
    .
  4. The vertical line test says that any given vertical line should only cross the graph
    .

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?