1. Relations
Topic 8.5.1

2. Relations 8.5.1 1.1.1 California Standards: What it means for you:
Lesson
1.1.1
Topic
8.5.1
Relations
California Standards:
16.0: Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.
17.0: Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.
What it means for you:
You’ll find out what relations are, and some different ways of showing relations.
Key words:
relation
ordered pair
domain
range

3. Topic
8.5.1
Lesson
1.1.1
Relations
Relations in Math are nothing to do with family members.
They’re useful for describing how the x and y values of coordinate pairs are linked.

4. 8.5.1 1.1.1 Relations A Relation is a Set of Ordered Pairs
Topic
8.5.1
Lesson
1.1.1
Relations
A Relation is a Set of Ordered Pairs
Before you can define a relation, you need to understand what “ordered pairs” are:
An ordered pair is just two numbers or letters, (or anything else) written in the form (x, y).
(12, 65)
(–3, d)
(–6, 5)
(3x, –9y)

5. Lesson
1.1.1
Topic
8.5.1
Relations
If x and y are both real numbers, ordered pairs can be plotted as points on a coordinate plane.
The first number in the ordered pair represents the x-coordinate.
The second number represents the y-coordinate.
For example, (–2, 1).

6. Relations 8.5.1 1.1.1 A relation is any set of ordered pairs.
Lesson
1.1.1
Topic
8.5.1
Relations
A relation is any set of ordered pairs.
Relations are represented using set notation, and can be named using a letter:
for example: m = {(1, 4), (2, 8), (3, 12), (4, 16)}.
Every relation has a domain and a range.
Domain: the set of all the first elements (x-values) of each ordered pair, for example: domain of m = {1, 2, 3, 4}
Range: the set of all the second elements (y-values) of each ordered pair, for example: range of m = {4, 8, 12, 16}

7. Lesson
1.1.1
Topic
8.5.1
Relations
An important point to note is that there may or may not be a reason for the pairing of the x and y values.
Looking at the relation m:
m = {(1, 4), (2, 8), (3, 12), (4, 16)}.
You can see that the x and y values are related by the equation y = 4x — but not all relations can be described by an equation.

8. Topic
8.5.1
Relations
Example 1
State the domain and range of the relation: r = {(1, 4), (3, 7), (3, 5), (5, 8), (9, 2)}.
Solution
Domain = {1, 3, 5, 9}
Range = {4, 7, 5, 8, 2}
Solution follows…

9. Relations 8.5.1 State the domain and range of the relation:
Topic
8.5.1
Relations
Example 2
State the domain and range of the relation:
f = {(a, 2), (b, 3), (c, 4), (d, 5)}.
Solution
Domain = {a, b, c, d}
Range = {2, 3, 4, 5}
Solution follows…

10. State the domain and range of each relation.
Lesson
1.1.1
Topic
8.5.1
Relations
Guided Practice
State the domain and range of each relation.
1. f (x) = {(1, 1), (–2, 1), (3, 5), (–3, 10), (–7, 12)}
2. f (x) = {(–1, –1), (2, 2), (3, –3), (–4, 4)}
3. f (x) = {(1, 2), (3, 4), (5, 6), (7, 8)}
Domain = {1, –2, 3, –3, –7} Range = {1, 5, 10, 12}
Domain = {–1, 2, 3, –4} Range = {–1, 2, –3, 4}
Domain = {1, 3, 5, 7} Range = {2, 4, 6, 8}
Solution follows…

11. State the domain and range of each relation.
Lesson
1.1.1
Topic
8.5.1
Relations
Guided Practice
State the domain and range of each relation.
4. f (x) = {(a, b), (c, d ), (e, f ), (g, h)}
5. f (x) = {(–1, 0), (–b, d ), (e, 3), (7, –f )}
6. f (x) = {(a, –a), (b, –b), (–c, c), (, –j )}
Domain = {a, c, e, g } Range = {b, d, f, h }
Domain = {–1, –b, e, 7} Range = {0, d, 3, –f } 1 2
Domain = {a, b, –c, } Range = {–a, –b, c, –j }
Solution follows…

12. Topic
8.5.1
Lesson
1.1.1
Relations
Mapping Diagrams Can Be Used to Represent Relations
One way to visualize a relation is to use a mapping diagram.
This mapping diagram represents the relation t = {(2, v), (3, c), (6, m)}. 2 3 6 c v m
In the diagram, the area on the left represents the domain.
The area on the right represents the range.
Domain
Range
The arrows show which member of the domain is paired with which member of the range.

13. 8.5.1 1.1.1 Relations Guided Practice 7. 8. 9. 10.
Topic
8.5.1
Lesson
1.1.1
Relations
Guided Practice
State the domain and range of each relation.
Domain = {2, 4, 6, 8, 10, 12}
Range = {4, 36, 100, 16, 64, 144}
Domain = {1, 2, –3, 0, –1}
Range = {a, b, c, d }
Domain = {a, e, i, o, u }
Range = {–8, –6, 5, 12}
Domain = {1, 2, 3, 4, 5}
Range = {a, b, c, d }
Solution follows…

14. Relations 8.5.1 1.1.1 You Can Use Input-Output Tables
Lesson
1.1.1
Topic
8.5.1
Relations
You Can Use Input-Output Tables
Input-output tables show relations.
The table below represents the relation {(1, 1), (2, 3), (3, 6), (4, 10), (5, 15)}.
The input is the domain and the output is the range.

15. Relations 8.5.1 1.1.1 Guided Practice
Lesson
1.1.1
Topic
8.5.1
Relations
Guided Practice
State the domain and range of each relation.
Domain = {1, 5, 12} Range = {32, 6, 0.3}
Domain = {–2, –1, 2, 3} Range = {8, 5, 13}
13.
Domain = {–3, –1, 0, 1} Range = {–26, 0, 1, 2}
Solution follows…

16. Relations 8.5.1 1.1.1 Relations Can Be Plotted as Graphs
Lesson
1.1.1
Topic
8.5.1
Relations
Relations Can Be Plotted as Graphs
Relations can be plotted on a coordinate plane, where the domain is represented on the x-axis and the range on the y-axis.
Graphs are most useful when you have continuous sets of values for the domain and range, so that you can connect points with a smooth curve or straight line.
This graph represents the relation {(x, y = x)} with domain {–2 ≤ x ≤ 8}.

17. Relations 8.5.1 1.1.1 Guided Practice
Lesson
1.1.1
Topic
8.5.1
Relations
Guided Practice
State the domain and range of each relation.
Domain = {–2 ≤ x ≤ 2} Range = {–3 ≤ y ≤ 3}
Domain = {–9 ≤ x ≤ 3} Range = {–3 ≤ y ≤ 9}
Solution follows…

18. Relations 8.5.1 Independent Practice
Topic
8.5.1
Relations
Independent Practice
Define each of the following terms.
1. Relation
2. Range
3. Domain
A set of ordered pairs, e.g., {(–1, 2), (0, 2), (1, 2), (0, 3)}.
The set of all second entries of the ordered pairs of a relation.
The set of all first entries of the ordered pairs of a relation.
Solution follows…

19. Relations 8.5.1 Independent Practice
Topic
8.5.1
Relations
Independent Practice
In Exercises 4–6, state the domain and range of each relation. 4. {(x, x2) : x  {–1, 2, 4}}
Domain = {–1, 2, 4} Range = {1, 4, 16}
Domain = {–2 ≤ x ≤ 2} Range = {–2 ≤ y ≤ 2}
Domain = {–1 ≤ x ≤ 1} Range = {–1 ≤ y ≤ 1}
Solution follows…

20. Relations 8.5.1 Independent Practice
Topic
8.5.1
Relations
Independent Practice
In Exercises 7–11, state the domain and range of each relation. 7.
8. f (x) = {(x, x2 – 1) : x Î {0, 1, 2}}
9. f (x) = {(x, –x2 + 3) : x Î {–2, –1, 0, 1, 2}}
Domain = {3, 4.5, 7} Range = {1, 2, 3}
Domain = {0, 1, 2} Range = {–1, 0, 3}
Domain = {–2, –1, 0, 1, 2} Range = {–1, 2, 3}
Solution follows…

21. Relations 8.5.1 Independent Practice x
Topic
8.5.1
Relations
Independent Practice
In Exercises 7–11, state the domain and range of each relation f (x) = {(x, ) : x Î {–1, 0, 2}}
11. f (x) = {(x, ) : x Î {–1, 0, 2, 3}}
x – 1 x
Domain = {–1, 0, 2} Range = {0, , 2} 1 2
x + 2
x + 1
Domain = {–1, 0, 2, 3} Range = {0, , , } 1 2 3 4 5
Solution follows…

22. Relations 8.5.1 Independent Practice
Topic
8.5.1
Relations
Independent Practice
In Exercises 12–14, state the domain and range of each relation.
Domain = {–3 ≤ x ≤ 3} Range = {–2 ≤ y ≤ 4}
Domain = {–4 ≤ x ≤ 4} Range = {–2 ≤ y ≤ 6}
Domain = {a, b, c, d, e} Range = {1, 2, 3, 4}
Domain = {1, –3, a, e} Range = {–1, 4, d, a, b}
Solution follows…

23. Topic
8.5.1
Relations
Round Up
The important thing to remember is that a relation is just a set of ordered pairs showing how a domain set and a range set are linked.