This document discusses functions, equations, and graphs. It defines relations and functions, and how to represent them using ordered pairs, mapping diagrams, and graphs. Functions are defined as relations where each input corresponds to exactly one output. The vertical line test can be used to determine if a graph represents a function. Function rules use function notation to represent the relationship between inputs and outputs algebraically. Examples are provided to illustrate these concepts.

1. CHAPTER 2: FUNCTIONS,
EQUATIONS, AND GRAPHS
2.1 Relations and Graphs

2. RELATIONS
 A relation is a set of pairs of input and output values.
 There are four ways to represent relations

3. EXAMPLE: REPRESENTING
RELATIONS
 The monthly average water temperature of the
Gulf of Mexico in Key West, Florida varies during
the year. In January, the average water
temperature is 69°F, in February, 70°F, in
March, 75°F and in April, 78°F. What is one way
to represent this relation?

4. DEFINITIONS
 The domain of a relation is the set of inputs,
also called the x-coordinates, of the ordered pairs.
 The range is the set of outputs, also called the y-
coordinates, of the ordered pairs.

5. EXAMPLE: FIND THE DOMAIN
AND RANGE OF EACH.

6. FUNCTIONS
 A function is a relation in which each element of
the domain corresponds to exactly one element of
the range.
 When using a mapping diagram to represent a
relation, a function has only one arrow from each
element of the domain.
 Example:

7. EXAMPLE: IS THE RELATION
A FUNCTION?

8. THE VERTICAL LINE TEST
 The vertical line test is used to test whether a
graph represents a function.
 The vertical line test states that if any vertical
line passes through more than one point on the
graph of a relation, then the relation is not a
function.

9. EXAMPLE: USE THE VERTICAL
LINE TEST. WHICH GRAPH(S)
REPRESENTS A FUNCTION?

10. FUNCTIONS
 A function rule is an equation that represents
an output value in terms of an input value.
 We write a function rule in function notation.

11. EXAMPLE: USING FUNCTION
NOTATION
 For f ( x ) = −2 x, + 5 what is the output for
1
the inputs: −3,0,&
4

12. EXAMPLE: USING FUNCTION
NOTATION
 For f ( x ) = −4 x + 1 ,what is the output for
the inputs: −2,0,5

13. EXAMPLE:
 Tickets to a concert are available online for $35 each
plus a handling fee of $2.50. The total cost is a
function of the number of tickets bought. What
function rule models the cost of the concert tickets?
Evaluate the function for 4 tickets.

14. EXAMPLE
 You are buying bottles of a sports drink for a
softball team. Each bottle costs $1.19. What
function rule models the cost of a purchase.
Evaluate the function for 15 bottles.