4-3 Relations

ObjectivesStudents will be able to:

1) Represent relations as sets of ordered pairs, tables, mappings, and graphs

2) Find the inverse of a relation

Terminology

• Relation: set of ordered pairs

• Domain: set of all x values in a relation

• Range: set of all y values in a relation

Ways to Represent a Relation1) As a set of ordered pairs

Example: {(1, 2), (-2, 4), (0, -3)}

2) As a table

3) As a graph

Example 1: Express each relation as a table, a graph, and a mapping. Then determine the domain and range.a) {(4, 3), (-2, -1), (-3, 3), (2, -4)}

Domain: Range:

You try.b) {(3, 2), (5, 2), (3, -1), (0, 1)}

Domain: Range:

Inverse: relation obtained from switching the coordinates of each ordered pair of the original relation

For example, if a relation is {(2, 1), (3, -5), (0,1)}, its inverse would be {(1, 2), (-5, 3), (1, 0)}.

Try and find the inverse of this relation. List the inverse as a set of ordered pairs.

4-6 Functions

ObjectivesStudents will be able to:

1) Determine whether a relation is a function2) Find functional values

Note: You cannot spell function without “fun”

Functions

• A function is a special type of relation in which each element of the domain is paired with exactly one element of the range.

• Let’s talk about what this means by looking at a real-life example of a relation. Let’s say that our domain is students, and our range is television shows. We can create a mapping of the relation.

• Let’s now recap using mathematical examples:

Example 2: Determine if the relation is a function.a) {(1, 3), (2, 3), (-1, 1)}yesb) {(1, 4), (2, 1), (1, 5)}No; the x value of 1 repeatsTry c) {(3, 1), (3, 2), (3, 4)} d) {(1, -1), (2, -1)}No

yes

Vertical Line Test

• When given a graph of a relation, one can perform a vertical line test to determine whether a relation is a function.

• If you drop in vertical lines, and they do not intersect the graph in more than one point, then the relation is a function. If they do intersect the graph in more than one point, then the relation is not a function.

Vertical Line Test

Example 3: Use the vertical line test to determine if the relation is a function.

a) b)

Not a functionYes, is a function

c) d)

yesno

Try these:e) f)

Not a functionYes, is a function

A further look at domain and range

• Remember that a domain is the set of our x values, and a range is the set of our y values.

• We can also determine the domain and range for linear equations, quadratic equations, absolute value equations, all types of equations.

Example 4: For each graph, determine the domain and range.a) Domain:

Range:

b) Domain:

Range:

c) Domain:

Range:

d) Domain:

Range:

Try these.e) f)

Domain: Domain:Range: Range:

Function Notation• Sometimes, an equation like might be written

as . This is what is referred to as function notation.

• The notation indicates that an equation is a function. In other words, if you graphed the equation, it would pass the vertical line test.

• Functions can be evaluated by taking the value in parenthesis and substituting each variable in the function with that value. After substitution is complete, simplify and combine any like terms.

a) f(-2) b) g(5) c) f(2d)

Try these.d) g(-4) e) f(3p) f) g(2a)