1. Understanding Functions
Lesson 6.1

2. Learning Objectives
I can tell whether a relation is a function.

3. Understanding Functions
A function is a special type of relation in which each input relates exactly to one output.
All functions are either one-to-one relations or many-to-one relations.
A function is a type of relation that assigns exactly one output to each input.

4. Examples of Real-World Situations Involving Functions

5. Examples of Real-World Situations Involving Functions

6. Examples of Real-World Situations Involving Functions

7. Examples of Real-World Situations Involving Functions
In the previous examples, each input produces exactly one output. Thus the three relations are all functions.

8. Examples of Real-World Situations That Are NOT Functions
When at least one input in a relation has more than one output, the relation is not a function.
For example, some libraries use the Dewey Decimal System to categorize books. All books about philosophy and psychology, for example, have a number in the 100's. You can think of the numbers in the following list as inputs. The outputs are the books that can be matched to those categories.

9. Examples of Real-World Situations That Are NOT Functions
You can see from the mapping diagram, for any given input, a category number, there may be more than one output, a book title.

10. Examples of Real-World Situations That Are NOT Functions
The mapping diagram shows a one-to-many relation. Because the inputs have more than one output, the relation is not a function.

11. Example 1 - Tell whether a relation is a function from a mapping diagram
The high jumpers at a track meet are wearing numbers on their uniforms. Each of the five high jumpers on the team made one jump. The height cleared by each athlete is shown in the table.

12. Example 1 - Tell whether a relation is a function from a mapping diagram
You can also write ordered pairs to show the relation between the athletes (represented by numbers) and the heights they cleared:
(1, 145) , (2, 143) , (3, 139) , (4, 151) , (5, 151)
Use a mapping diagram to represent the relation between the numbers of the athletes and the heights they cleared.

13. Example 1 - Tell whether a relation is a function from a mapping diagram
Tell whether the relation is a function and explain why.
SOLUTION:
The relation between the numbers of the athletes and the heights they cleared is a function because from the mapping diagram, each input is mapped to exactly one output. It is impossible for each high jumper to have two different recorded heights in one jump.

14. Example 1 - Tell whether a relation is a function from a mapping diagram
Suppose the inputs are the heights cleared by the athletes, and the outputs are the athletes' numbers. Use a mapping diagram to represent the relation. Is this relation a function?
The relation is not a function because one of the inputs has more than one output. From the mapping diagram, it is a one-to-many relation.

15. Guided Practice 1. 2.

16. Guided Practice 3.

17. Identify Functions Graphically
One way to see if a relation is a function is to use a mapping diagram. Another way is to use a graph.
For example, the mapping diagram shows a relation between a set of values x that are paired with a set of values y. The relation is a function because each input has exactly one output.

18. Identify Functions Graphically
You can also represent this function using a graph by writing and graphing ordered pairs (input x, output y) as points on a coordinate plane.
Notice that if you draw a vertical line through each point, each vertical line intersects exactly one point.

19. Identify Functions Graphically
Suppose there is another relation that is represented by the mapping diagram below.

20. Identify Functions Graphically
You can graph the relation by writing and graphing the ordered pairs. From the mapping diagram, the ordered pairs are:
(0, 1) , (1, 2) , (1, 4) , (2, 3) , (3, 4), and (4, 5)
As you can see, the graph of the relation does not pass the "vertical line test." So, this relation is not a function.

21. Tell whether a relation is a function from a graph.
The graph shows the relation between the heights eight students can jump into the air, y centimeters, and the students' heights, x centimeters. Tell whether the relation represented by the graph is a function.
From the graph, there is at least one vertical line that intersects the graph at more than one point. Based on the vertical line test, the relation is not a function.

22. Guided Practice
Tell whether the relation represented by the graph is a function. Explain.