How to Know if a Relation is a Function With or Without Graphing

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Easy homework help for functions in algebra or precalculus

Co-authored by Kyle Smith Reviewed by Joseph Meyer

Last Updated: March 2, 2023

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• Definitions
|
• Evaluate the Relation
|
• Vertical Line Test
|
• Tips

|Show more |Show less

Do you need to determine whether a relation is a function? The process may sound difficult, but it's actually pretty easy. To know if a relation is a function, just examine the inputs and outputs. When you’re given a set of ordered pairs, check whether any inputs have multiple outputs. If so, the relation is not a function. You can also do the vertical line test to check whether a relation is a function. This wikiHow guide shows you how to know when a relation is a function.

Things You Should Know

• Relations are a set of inputs and outputs, often shown as ordered pairs or a table.
• Functions are relations in which each input has exactly one output.
• Check whether each input has one output to know if a relation is a function.

Steps

Section 1 of 3:

Definitions

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1. 
   1
   Relations are a set of inputs and outputs. The inputs (domain) are paired with outputs (range).
2. 
   2
   Functions occur when each input has exactly one output. Different inputs can have the same output value. An input can not be assigned to multiple output values.

   • If you’re looking for quadratic function information, check out our guides on finding the inverse and finding the max or min.
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Section 2 of 3:

Evaluate the Relation

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1. 
   1
   Put your ordered pairs into a table. If you were given a set of ordered pairs, putting them into a table is helpful for checking if the relation is a function. Skip this step if you already have a table of inputs and outputs.

   • Take the left value (the x value) of each ordered pair and place them vertically in the left column (input) of a 2 column table.
   • Repeat for the right values (the y values), placing them in the right column (output).
2. 
   2
   Check whether any inputs have multiple outputs. If an input has multiple outputs, the relation is not a function.
3. 
   3
   Take a look at this non-function example. You’re given the following set of ordered pairs. Because the input 1 has two outputs, 2 and 7, this relation is not a function.

   • (1, 2)
   • (2, 4)
   • (3, 10)
   • (1, 7)
   • (5, 0)
4. 
   4
   Check out this function example. You’re given the following set of ordered pairs. Because each input only has one output, this relation is a function. Functions can have different inputs assigned to the same output. In this case, the inputs 3 and 4 have both been assigned to 10.

   • (1, 2)
   • (2, 4)
   • (3, 10)
   • (4, 10)
   • (5, 0)
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Section 3 of 3:

Vertical Line Test

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1. 
   1
   Plot the ordered pairs on a graph. Place a dot at each given ordered pair coordinate on an XY plane.
2. 
   2
   Move a vertical line across the graph. Use a straight-edge (like a ruler or another sheet of paper) to move a vertical line (parallel to the Y axis) across the entire graph. If the vertical line hits two or more points at the same X location, the relation is not a function.

   • In other words, if two or more points have the same X value and different Y values, the relation is not a function.
   • The relation is a function if you move the vertical line across the graph without hitting two or more points at the same time.
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Tips

• For general math tips, check out our guide on how to learn math.
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