1. 10
Quadratic Equations

2. 10.5 Introduction to Functions
Objectives
1. Understand the definition of a relation. 2. Understand the definition of a function. 3. Decide whether an equation defines a function. 4. Use function notation. 5. Apply the function concept in an application.

3. Understand the Definition of a Relation
In an ordered pair (x, y), x and y are called the components of the ordered pair.
Any set of ordered pairs is called a relation.
The set of all first components in the ordered pairs of a relation is the domain of the relation, and the set of all second components in the ordered pairs is the range of the relation.

4. Understand the Definition of a Relation
Example 1
Identify the domain and the range for the relation
{(–2,5), (1,6), (1,3), (3,3), (5,3)}.
The domain is the set of all first components, and the range is the set of all second components.
The domain is {–2, 1, 3, 5} and the range is {3, 5, 6}.
The relation in Example 1 is not a function, because the first
component, 1, corresponds to more than one second component.

5. Understand the Definition of a Function
A function is a set of ordered pairs in which each distinct first component corresponds to exactly one second component.

6. Understand the Definition of a Function
Example 2
Determine whether each relation is a function.
(a) {(1,2), (2,2), (3,4), (4,3), (5,6), (6,6)}
This relation IS a function because each number in the domain corresponds to only one number in the range.
(b) {(1,2), (2,2), (3,4), (4,3), (6,5), (6,6)}
This relation IS NOT a function because 6 in the domain corresponds to both 5 and 6 in the range.

7. Decide Whether an Equation Defines a Function
Vertical Line Test
If a vertical line intersects a graph in more than one point, the graph is not the graph of a function.

8. Decide Whether an Equation Defines a Function
Example 3
Decide whether each relation graphed or defined is a function.
(a) y = 3x – 2
Use the vertical line test. Any vertical line intersects the graph just once, so this is the graph of a function.

9. Decide Whether an Equation Defines a Function
Example 3 (concluded)
Decide whether each relation graphed or defined is a function.
(b) x2 + y2 = 4
The vertical line test shows that this is not the graph of a function; a vertical line could intersect the graph twice.

10. Use Function Notation
The letters f, g, and h are commonly used to name functions. For example, the function with defining equation y = 3x +5 may be written
f (x) = 3x + 5,
where f (x) is read “ f of x.” The notation f (x) is another way of writing y in a function.
For the function defined by f (x) = 3x + 5, if x = 7 then
f (7) = 3·7 + 5 = 26.
Read this result, f (7) = 26, as “ f of 7 equals 26.” The notation f (7) means the value of y when x is 7. The statement f (7) = 26 says that the value of y is 26 when x is 7. It also indicates the point (7, 26) lies on the graph of f.

11. Use Function Notation
CAUTION
The notation f (x) does not mean f times x. The symbol
f (x) means the value of the function f when evaluated
for x. It represents the y-value that corresponds to x.
Function Notation
In the notation f (x),
f is the name of the function,
x is the domain value,
and f (x) is the range value y for the domain value x.

12. Using Function Notation
Example 4
For the function defined by f (x) = –2x2 + 3x – 1, find the following.
(a) f (5)
(b) f (0)
Substitute 5 for x.
Substitute 0 for x.
f (5) = –2(5)2 + 3(5) – 1
f (0) = –2(0)2 + 3(0) – 1
f (5) = – – 1
f (0) = – 1
f (5) = –36

13. Apply the Function Concept in an Application
Example 5
The profits for Jeannie’s Jeans is given in the table below.
Year
2001
2002
2003
2004
2005
2006
Profit
(in millions)
$0.75
$1.6
$2.1
$3.8
$4.7
(a) If we choose years as the domain elements and profits as the range elements, does this relation represent a function? Why/why not?
Yes, this is a function because each year corresponds to exactly one profit.

14. Apply the Function Concept in an Application
Example 5 (concluded)
The profits for Jeannie’s Jeans is given in the table below.
Year
2001
2002
2003
2004
2005
2006
Profit
(in millions)
$0.75
$1.6
$2.1
$3.8
$4.7
(b) Find f (2002) and f (2005).
(c) For what x value does
f (x) equal $2.1 million?
f (2002) = $1.6 million
f (2003) = $2.1 million
f (2005) = $3.8 million