1.15: Algebraic Functions
What if you were given a set of x and y values? How could you determine whether the relation between those values represented a function? After completing this Concept you'll be able to analyze the domain and range of a relation to determine if it represents a function.
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CK12 Foundation: 0115S Relations and Functions
Guidance
A function is a special kind of relation. In a function, for each input there is exactly one output; in a relation, there can be more than one output for a given input.
Example A
Consider the relation that shows the heights of all students in a class. The domain is the set of people in the class and the range is the set of heights. This relation is a function because each person has exactly one height. If any person had more than one height, the relation would not be a function.
Notice that even though the same person can’t have more than one height, it’s okay for more than one person to have the same height. In a function, more than one input can have the same output, as long as more than one output never comes from the same input.
Example B
Determine if the relation is a function.
a) (1, 3), (1, 2), (3, 5), (2, 5), (3, 4)
b) (3, 20), (5, 25), (1, 5), (7, 12), (9, 2)
c)
Solution
The easiest way to figure out if a relation is a function is to look at all the
a) You can see that in this relation there are two different
b) Each value of
c) In this relation there are two different
When a relation is represented graphically, we can determine if it is a function by using the vertical line test. If you can draw a vertical line that crosses the graph in more than one place, then the relation is not a function.
Example C
For the following graphs, determine whether they are functions.
Solution:
1. Not a function. It fails the vertical line test.
2. A function. No vertical line will cross more than one point on the graph.
Watch this video for help with the Examples above.
CK12 Foundation: Relations and Functions
Vocabulary
 A function is a special kind of relation. In a function, for each input there is exactly one output; in a relation, there can be more than one output for a given input.
Guided Practice
For the following graphs, determine whether they are functions.
Solution:
1. A function. No vertical line will cross more than one point on the graph.
2. Not a function. It fails the vertical line test.
Practice
In 18, determine whether each relation is a function:
 (1, 7), (2, 7), (3, 8), (4, 8), (5, 9)
 (1, 1), (1, 1), (4, 2), (4, 2), (9, 3), (9, 3)

x−4−3−2−10y 169 410  (2, 6), (1, 3), (0, 0), (1, 3), (2, 6)
 (2, 8), (1, 1), (0, 0), (1, 1), (2, 8)
 (5, 10), (1, 5), (0, 10), (1, 5), (5, 10)

x0 1101001000y2−2 2−22 
Age2025253035Number of jobs by that age3 4 7 4 2
In 910, use the vertical line test to determine whether each relation is a function.
domain
The domain of a function is the set of values for which the function is defined.Range
The range of a function is the set of values for which the function is defined.Vertical Line Test
The vertical line test says that if a vertical line drawn anywhere through the graph of a relation intersects the relation in more than one location, then the relation is not a function.Image Attributions
Here you'll learn how to determine whether a relation is a function given its domain and range or its graph.
Concept Nodes:
domain
The domain of a function is the set of values for which the function is defined.Range
The range of a function is the set of values for which the function is defined.Vertical Line Test
The vertical line test says that if a vertical line drawn anywhere through the graph of a relation intersects the relation in more than one location, then the relation is not a function.