Understand the
Function Concept
Learn how a function assigns exactly one output to each input: the foundation of algebra, graphing, and real-world modeling.
8th Grade Math
What Is a Function?
A function is a rule that takes each input and assigns it to exactly one output. Think of it like a machine: you feed in a number, the machine does something to it, and one specific result comes out.
Inputs, Outputs, and Notation
Input (\( x \))
The value you put into the function. Also called the independent variable or domain element.
Output (\( y \))
The result that comes out. Also called the dependent variable or range element.
Notation
We write \( f(x) \) to mean "the output of function \( f \) when the input is \( x \)." So \( f(3) = 7 \) means input 3 gives output 7.
The Function Machine
The Formal Definition
This means three things must be true:
1
Every input has an output. No \( x \)-value in the domain is left unassigned.
2
Each input maps to only one output. If \( x = 4 \), then there is one and only one \( y \).
3
Multiple inputs can share the same output. Both \( f(2) = 5 \) and \( f(8) = 5 \) is perfectly fine.
Mapping Diagrams: Function vs Not a Function
Functions as Ordered Pairs
A function can be represented as a set of ordered pairs \( (x, y) \). The set is a function if no two pairs share the same first value with different second values.
✓ Function
\( \{(1,3),\; (2,5),\; (3,7),\; (4,9)\} \)
Each \( x \)-value appears once.
Each \( x \)-value appears once.
✕ Not a Function
\( \{(1,3),\; (2,5),\; (1,8),\; (4,9)\} \)
Input 1 maps to both 3 and 8.
Input 1 maps to both 3 and 8.
The Vertical Line Test
Function vs Relation
| Function | Relation | |
|---|---|---|
| Definition | Each input has exactly one output | Any set of ordered pairs |
| Input repeats? | Not allowed with different outputs | Allowed |
| Vertical Line Test | Passes (1 intersection) | May fail (2+ intersections) |
| Example | \( y = 2x + 1 \) | \( x^2 + y^2 = 25 \) |
How to Check for a Function
1
Identify the representation. Is it a table, a set of ordered pairs, a mapping diagram, an equation, or a graph?
2
Look at every input value. List all the \( x \)-values (or inputs from the left side of a mapping).
3
Check for repeats. Does any input appear more than once with a different output? If yes: not a function. If no: it is a function.
4
For graphs, use the Vertical Line Test. Sweep a vertical line left to right. If it ever crosses the graph twice, it is not a function.
Example 1: Table Check
Is this table a function?
| \( x \) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \( y \) | 4 | 7 | 10 | 13 | 16 |
Example 2: Ordered Pairs
Is this set of ordered pairs a function?
1
List the \( x \)-values: 2, 3, 2, 4.
2
Input \( x = 2 \) appears twice: once paired with 5 and once with 11.
3
Since the same input gives two different outputs, this is not a function.
Example 3: Graph with Vertical Line Test
Domain and Range
Domain
The set of all possible inputs (\( x \)-values). "What can I put in?"
Range
The set of all possible outputs (\( y \)-values). "What can come out?"
Common Mistakes to Avoid
✕ Same output means not a function
Two inputs can share the same output. \( f(2) = 5 \) and \( f(8) = 5 \) is perfectly fine. It is only a problem when the same input gives different outputs.
✕ Using a horizontal line test
The Vertical Line Test checks if a graph is a function. A horizontal line test checks for something else (one-to-one). Don't mix them up.
✕ Confusing equation with function
Not every equation is a function. \( x^2 + y^2 = 9 \) is a circle: for \( x = 0 \), both \( y = 3 \) and \( y = -3 \) are outputs. That fails the test.
Summary: Is It a Function?
| Representation | Function Test | Key Question |
|---|---|---|
| Ordered Pairs | No repeated \( x \) with different \( y \) | Does any \( x \) appear twice? |
| Table | Each \( x \)-column value is unique | Same input, two rows? |
| Mapping Diagram | Each input has one arrow | Does any input split? |
| Graph | Vertical Line Test | Does the line ever hit twice? |
| Equation | Solve for \( y \): one value per \( x \) | Can I get two \( y \)-values? |
Question 1 of 5
Which of the following sets of ordered pairs represents a function?
In choice C, every \( x \)-value (1, 2, 3, 5) is unique. The repeated \( y \)-value of 4 is allowed. Choices A, B, and D all have an \( x \)-value that repeats with different \( y \)-values.
Question 2 of 5
A graph fails the Vertical Line Test. What does this tell you?
If a vertical line crosses the graph at two or more points, then a single input produces multiple outputs. That violates the definition of a function.
Question 3 of 5
If \( f(x) = 3x - 2 \), what is \( f(5) \)?
Substitute \( x = 5 \): \( f(5) = 3(5) - 2 = 15 - 2 = 13 \). Function notation means "plug the input into the rule."
Question 4 of 5
In a mapping diagram, an input splits into two arrows pointing to different outputs. Is this a function?
A function requires each input to have exactly one output. If an input splits into two arrows, one input maps to two different outputs, which breaks the rule.
Question 5 of 5
Which equation does not represent \( y \) as a function of \( x \)?
The equation \( x^2 + y^2 = 16 \) is a circle. For many \( x \)-values (like \( x = 0 \)), there are two \( y \)-values (\( y = 4 \) and \( y = -4 \)). So it fails the function test. All the other choices produce exactly one \( y \) for each \( x \).