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Transcript
Chapter 3
Equations and
Inequalities in
Two Variables;
Functions
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
1
CHAPTER
3
Equations and Inequalities
in Two Variables;
Functions
3.2 The Slope of a Line
3.3 The Equation of a Line
3.5 Introduction to Functions and
Function Notation
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
2
3.5
Introduction to Functions and
Function Notation
1. Identify the domain and range of a relation
and determine whether a relation is a
function.
2. Find the value of a function.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
3
Relation: A set of ordered pairs.
Domain: The set containing initial values of a
relation; its input values; the first coordinates in
ordered pairs.
Range: The set containing all values that are paired to
domain values in a relation; its output values; the
second coordinates in ordered pairs.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
4
Function: A relation in which each value in the
domain is assigned to exactly one value in the range.
Domain
0
1
2
3
4
Range
2
4
6
8
10
Each element in the domain
has a single arrow pointing to
an element in the range.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
5
Every function is a relation, but not every relation is a
function.
If any value in the domain is assigned to more than one
value in the range, then the relation is not a function.
Domain
Range
0
2
1
4
2
6
10
12
not a function
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
6
Example
Identify the domain and range of the relation, then
determine if it is a function.
Birthdate
Family member
March 1
Donna
April 17
Dennis
Sept. 3
Catherine
October 9
Denise
Nancy
Domain:
{March
April 17,because
Sept 3, an
Oct 9}
The
relation
is not 1,
a function
element
the domain,
Sept.
3, is assigned
toNancy}
Range: in
{Donna,
Dennis,
Catherine,
Denise,
two names in the range.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
7
Determining the Domain and Range of a Graph
The domain is a set containing the first coordinate
(x-coordinate) of every point on the graph.
The range is a set containing the second coordinate
(y-coordinate) of every point on the graph.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
8
Vertical Line Test
To determine whether a graphical relation is a
function, draw or imagine vertical lines through each
value in the domain. If each vertical line intersects the
graph at only one point, the relation is a function. If
any vertical line intersects the graph more than once,
the relation is not a function.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
9
Example
For each graph, identify the domain and range. Then
state whether each relation is a function.
a.
b.
Domain: {x|x 1}
Not a function
Range: all real numbers
Domain: all real numbers
Function
Range: {y  1}
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
10
When written as an equation, the notation for a
function is a modification of an equation in two
variables.
y = 3x + 4 could be written as f(x) = 3x + 4
f(x) is read as “a function in terms of x” or “f of x”
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
11
Finding the Value of a Function
Given a function f(x), to find f(a), where a is a real
number in the domain of f, replace x in the function
with a and then evaluate or simplify.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
12
Example
For the function f(x) = 3x – 5, find the following.
a. f(2)
b. f(4)
c. f(a)
Solution
b. f(4) = 3x – 5 c. f(a) = 3x – 5
a. f(2) = 3x – 5
= 3(a) – 5
= 3(4) – 5
= 3(2) – 5
=6–5
= 3a – 5
= 12 – 5
=1
= 17
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
13
Example
Use the graph to find the indicated value of the function.
a. f(0)
b. f(2)
c. f(8)
10
Y
8
Solution
a. When x = 0, y = 0, so f(0) = 0.
6
4
2
-10 -8 -6
b. When x = 2, y = 2, so f(2) = 2.
-4 -2 0
-2
X
2
4
6
8 10
-4
-6
c. When x = 8, y = 4, so f(8) = 4.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
-8
-10
14