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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CK-12 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 values in the list or the table. If a value of appears more than once, and it’s paired up with different values, then the relation is not a function.
a) You can see that in this relation there are two different values paired with the value of 3. This means that this relation is not a function.
b) Each value of has exactly one value. The relation is a function.
c) In this relation there are two different values paired with the value of 2 and two values paired with the value of 1. The relation is not a function.
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.
CK-12 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 1-8, 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)
- (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)
In 9-10, 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
Description
Learning Objectives
Here you'll learn how to determine whether a relation is a function given its domain and range or its graph.