Transcript 2.2 Powerpoint

Chapter 2.2
Functions
Relations and Functions
Recall from Section 2.1 how we described one
quantity in terms of another.
The letter grade you receive in mathematics
course depends on your numerical score.
The amount you pay (in dollars) for gas at the gas
station depends on the number of gallons pumped.
The dollars spent on entertainment depends on the
type of entertainment.
We used ordered pairs to represent these
corresponding quantities. For example, (3, $5.25)
indicates you pay $5.25 for 3 gallons of gas.
Since the amount you pay depends on the number
of gallons pumped, the amount (in dollars) is
called the dependent variable and the number of
gallons pumped is called the independent variable.
Generalizing, if the value of the variable y
depends on the value of the variable x, then y is
the dependent variable and x is the independent
variable.
dependent variable
independen t variable
(x, y)
Because we can write related quantities using
ordered pairs, a set of ordered pairs such as
{(3, 5.25), (8, 10), (10, 12.50)}
is called a relation.
Relation
A relation is a set of ordered pairs.
Function
A function is a relation in which, for each value of
the first component of the ordered pairs, there is
exactly one value of the second component.
Note
The relation from the beginning of this section
representing the number of gallons of gasoline and
the corresponding cost is a function such each xvalue is paired exactly one y-value.
You would not be happy, for example, if you and a
friend each pumped 20 gallons of regular gasoline
a the same station and you bill was $32 while his
bill was $28.
Example 1 Deciding Whether Relations Define Functions
Decide whether each relation defines a function.
F = {(1, 2), (-2, 4), (3, -1)}
Example 1 Deciding Whether Relations Define Functions
Decide whether each relation defines a function.
G = {(1, 1), (1, 2), (2, 3)}
Example 1 Deciding Whether Relations Define Functions
Decide whether each relation defines a function.
H = {(-4, 1), (-2, 1), (-2, 0)}
In a function there is exactly one value for the
dependent variable, the second component, for
each value of the independent variable, the first
component. This is what makes functions so
important in applications.
Relations and functions can
also be expressed as a
correspondence or mapping
from one set to another, as
shown in figure 17 for
function F and relation H
from Example 1.
The arrow from 1 to 2
indicates that the ordered
pair (1, 2) belongs to F—
each first component is
paired with exactly one
second component.
In the mapping for relation
H, which is not a function,
the first component -2 is
paired with two different
second components, 1 and
0.
Since relations and functions are sets of ordered
pairs, we can represent them using tables and
graphs.
y
A table and graph for
function F is shown in
figure 18.
x
y
1
2
-2
4
(-2, 4)
(1, 2)
3 -1
x
(3, -1)
Graph of F
Figure 18
Finally, we can describe a relation or function
using a rule that tells how to determine the
dependent variable for a specific value of the
independent variable.
The rule may be given in words: for instance, “the
dependent variable is twice the independent
variable.”
Usually the rule is an equation:
dependent variable
independen t variable
y  2x
Note
Another way to think of a function relationship is
to think of the independent variable as an input
and the dependent variable as an output.
This is illustrated by the input-output (function)
machine for the function defined by
y  2x
Domain and Range
For every relation there are two important set of
elements called the domain and the range
Domain and Range
In a relation, the set of all values of the
independent variable (x) is the domain; the set of
all values of the dependent variable (y) is the
range
Example 2 Finding Domains and Ranges of Relations
Give the domain and range of each relation. Tell
whether the relations defines a function.
{(3, -1), (4, 2 ), (4, 5), (8, 8)}
Example 2 Finding Domains and Ranges of Relations
Give the domain and range of each relation. Tell
whether the relations defines a function.
(b)
4
A
6
B
7
-3
C
Example 2 Finding Domains and Ranges of Relations
Give the domain and range of each relation. Tell
whether the relations defines a function.
(c)
x
y
-5
2
0
2
5
2
Example 3 Finding Domains and Ranges of Relations
Give the domain and range of each relation.
(a)
(-1, 1)
(1, 2)
(0, -1)
(4, -3)
Example 3 Finding Domains and Ranges of Relations
Give the domain and range of each relation.
Domain
(b)
Range
Example 3 Finding Domains and Ranges of Relations
Give the domain and range of each relation.
(c)
Example 3 Finding Domains and Ranges of Relations
Give the domain and range of each relation.
(d)
Since relations are defined by equations, such as
y = 2x + 3 and y2 = x, we must sometimes
determine the domain of a relation from its
equation. In this book, we assume the following
agreement on the domain of a relation.
To illustrate this agreement, since any real number
can be used as a replacement for x in y = 2x + 3,
the domain of this function is the set of all real
numbers.
As another example, the function defined by
y = 1/x has all real numbers except 0 as domain,
since y is undefined if x = 0.
In general, the domain of a function defined by an
algebraic expression is all real numbers, except
those numbers that lead to division by 0 or an
even root of a negative number.
Determining Functions from Graphs or Equations
Most of the relations we have seen in the
examples are functions—that is, each x-value
corresponds to exactly one y-value. Since each
value of x leads to only one value of y in a
function, any vertical line drawn through the
graph of a function must intersect the graph in at
most one point. This is the vertical line test for a
function
The graph in Figure 19(a) represents a function
Example 4 Using the Vertical Line Test
Use the vertical line test to determine whether
each relation is a function.
(a)
(-1, 1)
(1, 2)
(0, -1)
(4, -3)
Example 4 Using the Vertical Line Test
Use the vertical line test to determine whether
each relation is a function.
Domain
(b)
Range
Example 4 Using the Vertical Line Test
Use the vertical line test to determine whether
each relation is a function.
(c)
Example 4 Using the Vertical Line Test
Use the vertical line test to determine whether
each relation is a function.
(d)
Example 5 Identifying Functions Domains and Ranges From Equations
Decide whether each relation defines a function and
give the domain and range.
y=x+4
Example 5 Identifying Functions Domains and Ranges From Equations
Decide whether each relation defines a function
and give the domain and range.
y  2x 1
Example 5 Identifying Functions Domains and Ranges From Equations
Decide whether each relation defines a function
and give the domain and range.
Example 5 Identifying Functions Domains and Ranges From Equations
Decide whether each relation defines a function
and give the domain and range.
2
y
=x
Example 5 Identifying Functions Domains and Ranges From Equations
Decide whether each relation defines a function
and give the domain and range.
Example 5 Identifying Functions Domains and Ranges From Equations
Decide whether each relation defines a function
and give the domain and range.
y  x 1
Example 5 Identifying Functions Domains and Ranges From Equations
Decide whether each relation defines a function
and give the domain and range.
5
y
x 1
Example 5 Identifying Functions Domains and Ranges From Equations
Decide whether each relation defines a function
and give the domain and range.
Function Notation
When a function f is defined with a rule or an
equation using x and y for the independent and
dependent variables, we say “y is a function of x”
to emphasize that y depends on x.
We use the notation
y = f(x)
called function notation, to express this and read
f(x) as “f of x.”
For example if y = f(x) = 9x – 5 and x = 2, then
we find y, or f(2), by replacing x with 2.
y = f(x) = 9x - 5
= 9(2) – 5
= 18 – 5
= 13
if y = f(x) = 9x - 5
f(0) =
if y = f(x) = 9x - 5
f(-3) =
Example 6 Using Function Notation
Let f(x) = -x2 + 5x -3
Find f(2)
Example 6 Using Function Notation
Let f(x) = -x2 + 5x -3
Find f(q)
Example 7 Using Function Notation
Let g(x) = 2x + 3
Find g(a+1)
Example 8 Using Function Notation
Let f(x) = 3x - 7
Find f(3)
Example 8 Using Function Notation
f = {(-3,5 ), (0,3), (3,1), (6,-1),
Find f(3)
Example 8 Using Function Notation
Find f(3)
Example 9 Using Function Notation
Find f(3)
Example 9 Using Function Notation
Find f(3)
Example 9 Wrting Equations Using Function Notation
Rewrite each equation using function notation.
Then find f(-2) and find f(a)
y = x2 + 1
Example 9 Wrting Equations Using Function Notation
Rewrite each equation using function notation.
Then find f(-2) and find f(a)
x – 4y = 5
Increasing, Decreasing, and Constant Functions
Informally speaking, a function increases on an
interval of its domain if its graph rises from left to
right on the interval.
It decreases on an interval of its domain if its
graph falls from left to right on the interval.
It is constant on an interval of its domain if its
graph is horizontal on the interval.
Example 2 Finding Domains and Ranges of Relations
In Figure 24 the function increases on the interval
[-2, 1] because the y-values continue to get larger
in that interval.
Example 2 Finding Domains and Ranges of Relations
It is constant on the interval [1, 4] because the yvalues are always 5 for all x-values there.
Example 2 Finding Domains and Ranges of Relations
In Figure 24 the function decreases on the interval
[4, 6] because the y-values continue to get smaller
in that interval.
Example 10 Determinng Intervals over Which a Function is Increasing,
Decreasing, or Constant
Determine the intervals over which a function is
increasing, decreasing, or constant
Example 11 Interpreting a Graph
Figure 27 shows the relationship
between the number of gallons of water
in a small swimming pool and the time
in hours.
By looking at this graph of the function,
we can answer questions about the
water level in the pool at various times.
Example 11 Interpreting a Graph
For example, at the
time 0 the pool is
empty.
Example 11 Interpreting a Graph
The water level then
increases,
stays constant for a
while,
decreases,
then becomes
constant again.
Example 11 Interpreting a Graph
What is the
maximum number
of gallons of water
in the pool?
When is the
maximum water
level first reached?
Example 11 Interpreting a Graph
For how long is the
water level
increasing?
decreasing?
constant?
Example 11 Interpreting a Graph
How many gallons
of water are in the
pool after 90 hours?
Example 11 Interpreting a Graph
Describe a series of
events that could
account for the
water level changes
shown in the graph?