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1. RELATIONS AND FUNCTIONS

2. VOCABULARY
RELATION: A relation is a set of
ordered pairs.

3. VOCABULARY
DOMAIN: The domain of a relation is
the set of all inputs.

4. VOCABULARY
RANGE: The range of a relation is the
set of all outputs.

5. VOCABULARY
INPUT: Each number in a domain is an
input.

6. VOCABULARY
OUTPUT: Each number in a range is an
output.

7. VOCABULARY
FUNCTION: A function is a relation
with the property that for each input
there is exactly one output.

8. In order to be a function, every X must have a one Y.
No Ys can share any Xs.

9. Think about airplanes. Lots of planes fly at the same HEIGHT (y value).
But generally, we try to make sure that planes do not fly on top of each
other (the same x value).
Why do we make sure planes do not occupy the same x value?

10. Sure. Cool.
Great.
No.

11. VOCABULARY
VERTICAL LINE TEST: The vertical line test says that if you can find a
vertical line passing through more than one point of a graph of a relation,
then the relation is not a function. Otherwise, the relation is a function.

12. EXAMPLE 1—Identifying the Domain and Range
Identify the domain and range of the relation represented by the table
below that shows one Norway Spruce tree’s height at different ages.

13. EXAMPLE 1—Identifying the Domain and Range
The relation consists of the ordered pairs _____, _____, _____,
_____, and _____.

14. EXAMPLE 1—Identifying the Domain and Range

15. EXAMPLE 1—Identifying the Domain and Range
The domain of the relation is the set of all ______, or
______________.

16. EXAMPLE 1—Identifying the Domain and Range
The domain of the relation is the set of all INPUTS, or
X-COORDINATES.

17. EXAMPLE 1—Identifying the Domain and Range
The range is the set of all _______, or _____________.

18. The range is the set of all OUTPUTS, or Y-
COORDINATES.

19. EXAMPLE 1—Identifying the Domain and Range
So, what would be the domain and range of our previous
data sets?
DOMAIN: __, __, __, __, __
RANGE: __, __, __, __, __

20. EXAMPLE 1—Identifying the Domain and Range
DOMAIN: 5, 10, 15, 20, 25
RANGE: 13, 25, 34, 43, 52

21. EXAMPLE 2—Representing a Relation
Represent a relation (−3, 2), (−2, −2), (1, 1), (1, 3), (2, −3) as indicated.
Graph the ordered pairs as POINTS in a coordinate plane.

22. EXAMPLE 2—Representing a Relation
Graph the ordered pairs as POINTS in a coordinate plane.
Represent a relation (−3, 2), (−2, −2), (1, 1), (1, 3), (2, −3) as indicated.

23. EXAMPLE 2—Representing a Relation
List the inputs and the outputs in order. Draw arrows from
the INPUTS to their OUPUTS.
Represent a relation (−3, 2), (−2, −2), (1, 1), (1, 3), (2, −3) as indicated.

24. EXAMPLE 2—Representing a Relation
List the inputs and the outputs in order. Draw arrows from
the INPUTS to their OUPUTS.
Represent a relation (−3, 2), (−2, −2), (1, 1), (1, 3), (2, −3) as indicated.

25. EXAMPLE 3—Identifying Functions
Tell whether the relation is a function.
The relation in Example 1.
The relations __ a function because
___________________________ ___________. This makes
sense, as a single tree can have ONLY ONE height at a given
point in time.

26. EXAMPLE 3—Identifying Functions
Tell whether the relation is a function.
The relation in Example 1.
The relation IS a function because EVERY INPUT IS
PAIRED WITH EXACTLY ONE OUTPUT. This makes
sense, as a single tree can have ONLY ONE height at a
given point in time.

27. EXAMPLE 3—Identifying Functions
Tell whether the relation is a function.
The relation in Example 2.
The relation _____ a function because
___________________
_________________________.

28. EXAMPLE 3—Identifying Functions
Tell whether the relation is a function.
The relation in Example 2.
The relation IS NOT a function because THE INPUT
1 IS PAIRED WITH TWO OUTPUTS, 1 AND 3.

29. CHECKPOINT—Identify the domain and range of the relation and tell
whether the relation is a function.
(−5, 2), (−3, −1), (−1, 0), (2, 3), (5, 4) (−4, −3), (−3, 2), (0, 0), (1, −1), (2, 3), (3, 1), (3, −2)
1 2

30. EXAMPLE 4—Using the Vertical Line Test
In the graph, no vertical line passes through more than one point.
So, the relation represented by the graph ________________.

31. EXAMPLE 4—Using the Vertical Line Test
In the graph, no vertical line passes through more than one point.
So, the relation represented by the graph IS A FUNCTION.

32. EXAMPLE 4—Using the Vertical Line Test
In the graph, the vertical line shown passes through two points. So, the
relation represented by the graph ____________________.

33. EXAMPLE 4—Using the Vertical Line Test
In the graph, the vertical line shown passes through two points. So, the
relation represented by the graph IS NOT A FUNCTION.

34. 3
State whether the following relations are
functions.
4
5 6
(3, 6), (2, 4), (3, 7), (-7, 9) (1, 2), (3, 4), (5, 6), (7, 8)