1 topic 8.5.1 relations. 2 lesson 1.1.1 california standards: 16.0: students understand the concepts...

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Topic 8.5.1Topic 8.5.1

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Lesson

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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.

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Key words:• relation• ordered pair• domain• range

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Relations in Math are nothing to do with family members.

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They’re useful for describing how the x and y values of coordinate pairs are linked.

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A Relation is a Set of Ordered Pairs

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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).

(–6, 5) (12, 65) (3x, –9y)(–3, d)

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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).

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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}

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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:

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.

m = {(1, 4), (2, 8), (3, 12), (4, 16)}.

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Example 1

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Solution

State the domain and range of the relation:

r = {(1, 4), (3, 7), (3, 5), (5, 8), (9, 2)}.

Solution follows…

Domain = {1, 3, 5, 9}

Range = {4, 7, 5, 8, 2}

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Example 2

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Solution

State the domain and range of the relation:

f = {(a, 2), (b, 3), (c, 4), (d, 5)}.

Solution follows…

Domain = {a, b, c, d}

Range = {2, 3, 4, 5}

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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)}

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Guided Practice

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Solution follows…

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}

State the domain and range of each relation.

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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 )}

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Guided Practice

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Solution follows…

Domain = {a, b, –c, } Range = {–a, –b, c, –j }

Domain = {a, c, e, g } Range = {b, d, f, h }

Domain = {–1, –b, e, 7} Range = {0, d, 3, –f }

State the domain and range of each relation.

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Mapping Diagrams Can Be Used to Represent Relations

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One way to visualize a relation is to use a mapping diagram.

In the diagram, the area on the left represents the domain.

The area on the right represents the range.

Range Domain

This mapping diagram represents the relation t = {(2, v), (3, c), (6, m)}. 2

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6

c

v

m

The arrows show which member of the domain is paired with which member of the range.

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State the domain and range of each relation.

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Guided Practice

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Solution follows…

7. 8.

9. 10.

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 }

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You Can Use Input-Output Tables

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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.

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State the domain and range of each relation.

11. 12.

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Guided Practice

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Solution follows…

Domain = {1, 5, 12}Range = {32, 6, 0.3} Domain = {–2, –1, 2, 3}

Range = {8, 5, 13}

Domain = {–3, –1, 0, 1}Range = {–26, 0, 1, 2}

13.

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Relations Can Be Plotted as Graphs

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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}.

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State the domain and range of each relation.

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Guided Practice

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Solution follows…

Domain = {–2 ≤ x ≤ 2}Range = {–3 ≤ y ≤ 3}

Domain = {–9 ≤ x ≤ 3}Range = {–3 ≤ y ≤ 9}

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Independent Practice

Solution follows…

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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.

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Independent Practice

Solution follows…

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Domain = {–2 ≤ x ≤ 2} Range = {–2 ≤ y ≤ 2}

Domain = {–1, 2, 4} Range = {1, 4, 16}

Domain = {–1 ≤ x ≤ 1} Range = {–1 ≤ y ≤ 1}

In Exercises 4–6, state the domain and range of each relation.

4. {(x, x2) : x {–1, 2, 4}}

5. 6.

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In Exercises 7–11, state the domain and range of each relation.

7.

8. f (x) = {(x, x2 – 1) : x {0, 1, 2}}

9. f (x) = {(x, –x2 + 3) : x {–2, –1, 0, 1, 2}}

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Independent Practice

Solution follows…

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Domain = {0, 1, 2} Range = {–1, 0, 3}

Domain = {3, 4.5, 7} Range = {1, 2, 3}

Domain = {–2, –1, 0, 1, 2} Range = {–1, 2, 3}

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In Exercises 7–11, state the domain and range of each relation.

10. f (x) = {(x, ) : x {–1, 0, 2}}

11. f (x) = {(x, ) : x {–1, 0, 2, 3}}

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Independent Practice

Solution follows…

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Domain = {–1, 0, 2} Range = {0, , 2}12

Domain = {–1, 0, 2, 3} Range = {0, , , }12

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45

x – 1x

x + 2x + 1

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Independent Practice

Solution follows…

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In Exercises 12–14, state the domain and range of each relation.

12. 13.

14. 15.

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}

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Round UpRound Up

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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.