1. Section 1.2 Basics of Functions

2. Definition of a Relation
A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation and the set of all the second components is called the range of the relation.
Domain:
{sitting, walking, aerobics, tennis, running, swimming}
Definition of a Relation
Range:
{ 80, 325, 505, 720, 790}
You do not have to list 505 twice
Example
Find the domain and the range.
Domain:
{98.3, 98.6, 99.1}
Range:
{ Felicia, Gabriella, Lakeshia}

3. This is called a mapping because it maps one element to another.
A relation in which each member of the domain corresponds to EXACTLY one member of the range is a function. Notice that more than one element in the domain can correspond to the same element in the range. Aerobics AND tennis both burn 505 calories per hour.
Functions
This is called a mapping because it maps one element to another.

4. Is this a function?
Does each member of the domain correspond to precisely one member of the range?
This relation is not a function because there is a member of the domain that corresponds to two members of the range.
505 corresponds to aerobics and tennis.

5. Definition of a Function
A Function is a correspondence from a first set, called the domain, to a second set called the range, such that each element in the domain corresponds to exactly one element in the range.

6. Determine whether each relation is a function?
Example
Determine whether each relation is a function?
yes
No 2 is repeated as an x-value
yes

7. Functions as Equations
Example
Not every set of ordered pairs defines a function. Not all equations with variables x and y define a function. If an equation is solved for y and more than one value of y can be obtained for a given x, then the equation does not define y as a function of x. Thus the equation is not a function.
Determine whether each equation defines y as a function of x.
Functions as Equations
yes
yes no

8. Functions as Equations
Here is an equation that models paid vacation days each year as a function of years working for the company.
The variable x represents years working for a company. The variable y represents the average number of vacation days each year.
The variable y is a function of the variable x.
For each value of x, there is one and only one value of y.
The variable x is called the independent variable because it can be assigned any value from the domain. Thus x can be assigned any positive integer representing the number of years working for a company.
The variable y is called the dependent variable because its value depends on x. Paid vacations days depend on the number of years working for a company.
Functions as Equations

9. Function Notation
The special notation f(x), read “f of x” or “f at x” represents the value of the function at the number x.
If a function f and x represents the independent variable, the notation f(x) corresponds to the y-value for a given x.
We are evaluation the function at 10 when we substitute 10 for x.
What’s the answer?

10. Evaluate each of the following.
Example
Evaluate each of the following. no

11. Evaluate each of the following.
Example
Evaluate each of the following. no

12. Graphs of Functions x f(x) = -2x (x, y) g(x) = -2x+3 -2 f(-2) = 4
The graph of a function is the graph of its ordered pairs. First find the ordered pairs, then graph the functions. Graph the functions: x
f(x) = -2x
(x, y)
g(x) = -2x+3 -2
f(-2) = 4
(-2, 4)
g(-2) = 7
(-2, 7) -1
f(-1) = 2
(-1, 2)
g(-1) = 5
(-1, 5)
f(0) = 0
(0, 0)
g(0) = 3
(0, 3) 1
f(1) = -2
(1, -2)
g(1) = 1
(1, 1) 2
f(2) = -4
(2, -4)
g(2) = -1
(2, -1)
Graphs of Functions
See the next slide.

13. g(x)
f(x)

14. Example
Graph the following functions f(x) = 3x-1 and g(x) = 3x
g(x)
f(x)

15. If any vertical line intersects a graph in more than one point, the graph does NOT define y as a function of x.
The Vertical Line Test
The first graph is a function, the second is not.

16. Example
Use the vertical line test to identify graphs in which y is a function of x.

17. Example
Use the vertical line test to identify graphs in which y is a function of x.

18. Information from Graphs
You can obtain information about a function from its graph. A closed dot indicates that the graph does not extend beyond this point, and the point belongs to the graph. An open dot indicates that the graph does not extend beyond this point and the point does not belong to the graph (hole). An arrow indicates that the graph extends indefinitely in the direction in which the arrow points.
Information from Graphs

19. Example
Analyze the graph.

20. Identify the function’s domain and range from the graph.

21. Identify the Domain and Range from the graph.
Example
Identify the Domain and Range from the graph.

22. Identify the Domain and Range from the graph.
Example
Identify the Domain and Range from the graph.

23. Identify the Domain and Range from the graph.
Example
Identify the Domain and Range from the graph.

24. Identifying Intercepts from Graphs
To find the x-intercepts, look for points at which the graph crosses the x-axis.
To find the y-intercepts, look for points at which the graph crosses the y-axis.
The zeros of a function, f, are the x values for which f(x) = 0. These are the x-intercepts.
Identifying Intercepts from Graphs

25. Example
Find the x intercept(s). Find f(-4)

26. Example
Find the y intercept. Find f(2)

27. Example
Find the x and y intercepts. Find f(5).

28. 1. Find f(7).
(a)
(b)
(c)
(d)
Exit Ticket d

29. 2. Find the Domain and Range.
(b)
(c)
(d)
Exit Ticket c

30. (a)
(b)
(c)
(d) a
Exit Ticket