Transcript Functions - SaigonTech

10TH
EDITION
COLLEGE
ALGEBRA
LIAL
HORNSBY
SCHNEIDER
2.3 - 1
2.3 Functions
Relations and Functions
Domain and Range
Determining Functions from Graphs or
Equations
Function Notation
Increasing, Decreasing, and Constant
Functions
2.3 - 2
Relation
A relation is a set of ordered pairs.
2.3 - 3
Function
A function is a relation in which, for
each distinct value of the first
component of the ordered pair, there
is exactly one value of the second
component.
2.3 - 4
Motion Problems
Note The relation from the beginning
of this section representing the number of
gallons of gasoline and the corresponding
cost is a function since each x-value is
paired with exactly one y-value. You would
not be happy if you and a friend each
pumped 10 gal of regular gasoline at the
same station and your bills were different.
2.3 - 5
Example 1
DECIDING WHETHER
RELATIONS DEFINE FUNCTIONS
Decide whether the relation defines a
function.
F  (1,2),( 2,4)(3, 1)
Solution Relation F is a function, because
for each different x-value there is exactly
one y-value. We can show this
correspondence as follows.
1,  2, 3
x-values of F
2, 4, 1
y-values of F
2.3 - 6
Example 1
DECIDING WHETHER
RELATIONS DEFINE FUNCTIONS
Decide whether the relation defines a
function.
G  (1,1),(1,2)(1,3)(2,3)
Solution As the correspondence shows
below, relation G is not a function because
one first component corresponds to more
than one second component.
1, 2
x-values of G
1, 2, 3
y-values of G
2.3 - 7
Example 1
DECIDING WHETHER
RELATIONS DEFINE FUNCTIONS
Decide whether the relation defines a
function.
H  (  4,1),( 2,1)( 2,0)
Solution In relation H the last two ordered
pairs have the same x-value paired with two
different y-values, so H is a relation but not a
function.
Different y-values
H  (  4,1), ( 2,1)( 2,0)
Not a function
Same x-values
2.3 - 8
Mapping
Relations and functions can also be expressed as
a correspondence or mapping from one set to
another. In the example below the arrows from 1
to 2 indicates that the ordered pair (1, 2) belongs
to F. Each first component is paired with exactly
one second component.
x-axis values
y-axis values
1
–2
3
2
4
–1
2.3 - 9
Mapping
In the mapping for relations H, which is not a
function, the first component –2 is paired with two
different second components, 1 and 0.
x-axis values
y-axis values
–4
–2
1
0
2.3 - 10
Relations
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.
2.3 - 11
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.
2.3 - 12
Example 2
FINDING DOMAINS AND RANGES
OF RELATIONS
Give the domain and range of the relation.
Tell whether the relation defines a function.
a. (3, 1),(4,2),(4,5),(6,8)
The domain, the set of x-values, is {3, 4, 6};
the range, the set of y-values is {–1, 2, 5, 8}.
This relation is not a function because the
same x-value, 4, is paired with two different
y-values, 2 and 5.
2.3 - 13
Example 2
FINDING DOMAINS AND RANGES
OF RELATIONS
Give the domain and range of the relation.
Tell whether the relation defines a function.
b.
4
6
7
–3
100
200
300
The domain is {4, 6, 7, –3}; the range is
{100, 200, 300}. This mapping defines a
function. Each x-value corresponds to
exactly one y-value.
2.3 - 14
Example 2
FINDING DOMAINS AND RANGES
OF RELATIONS
Give the domain and range of the relation.
Tell whether the relation defines a function.
c.
x
–5
0
5
y
2
2
2
This relation is a set of
ordered pairs, so the
domain is the set of xvalues {–5, 0, 5} and the
range is the set of y-values
{2}. The table defines a
function because each
different x-value
corresponds to exactly one
y-value.
2.3 - 15
Example 3
FINDING DOMAINS AND RANGES
FROM GRAPHS
Give the domain and range of each relation.
y
a.
(1, 2)
(– 1, 1)
x
(0, – 1)
The domain is the set
of x-values which are
{– 1, 0, 1, 4}.
The range is the set of
y-values which are
{– 3, – 1, 1, 2}.
(4, – 3)
2.3 - 16
FINDING DOMAINS AND RANGES
FROM GRAPHS
Example 3
Give the domain and range of each relation.
y
b.
6
x
–4
4
The x-values of the
points on the graph
include all numbers
between –4 and 4,
inclusive. The yvalues include all
numbers between –6
and 6, inclusive.
The domain is [–4, 4].
The range is [–6, 6].
–6
2.3 - 17
FINDING DOMAINS AND RANGES
FROM GRAPHS
Example 3
Give the domain and range of each relation.
y
c.
x
The arrowheads
indicate that the line
extends indefinitely
left and right, as well
as up and down.
Therefore, both the
domain and the range
include all real
numbers, written
(– , ).
2.3 - 18
FINDING DOMAINS AND RANGES
FROM GRAPHS
Example 3
Give the domain and range of each relation.
y
d.
The arrowheads indicate
that the line extends
indefinitely left and right,
as well as upward. The
domain is (– , ).
Because there is at least
x y-value, –3, the range
includes all numbers
greater than, or equal to
–3 or [–3, ).
2.3 - 19
Agreement on Domain
Unless specified otherwise, the domain
of a relation is assumed to be all real
numbers that produce real numbers
when substituted for the independent
variable.
2.3 - 20
Vertical Line Test
If each vertical line intersects a graph
in at most one point, then the graph is
that of a function.
2.3 - 21
Example 4
USING THE VERTICAL LINE TEST
Use the vertical line test to determine whether each
relation graphed is a function.
y
a.
(1, 2)
(– 1, 1)
This graph
represents a
x function.
(0, – 1)
(4, – 3)
2.3 - 22
USING THE VERTICAL LINE TEST
Example 4
Use the vertical line test to determine whether each
relation graphed is a function.
y
b.
6
x
–4
4
This graph fails the
vertical line test, since
the same x-value
corresponds to two
different y-values;
therefore, it is not the
graph of a function.
–6
2.3 - 23
USING THE VERTICAL LINE TEST
Example 4
Use the vertical line test to determine whether each
relation graphed is a function.
y
c.
x
This graph
represents a
function.
2.3 - 24
USING THE VERTICAL LINE TEST
Example 4
Use the vertical line test to determine whether each
relation graphed is a function.
y
d.
x
This graph
represents a
function.
2.3 - 25
Relations
Note Graphs that do not represent
functions are still relations. Remember that
all equations and graphs represent
relations and that all relations have a
domain and range.
2.3 - 26
IDENTIFYING FUNCTIONS,
DOMAINS, AND RANGES
Decide whether each relation defines a function and
give the domain and range.
Example 5
a. y  x  4
Solution Since y is always found by adding 4 to x,
each value of x corresponds to just one value of y
and the relation defines a function.
x can be any real number, so the domain is
x  x is a real number or
(  ,  ).
Since y is always 4 more than x, y may also be any
real number, and so the range is (  ,  ).
2.3 - 27
IDENTIFYING FUNCTIONS,
DOMAINS, AND RANGES
Decide whether each relation defines a function and
give the domain and range.
Example 5
b. y  2x  1
Solution For any choice of x in the domain, there
is exactly one corresponding value for y (the radical
is a nonnegative number), so this equation is a
function.
Since the equation involves a square root, the
quantity under the radical cannot be negative.
2.3 - 28
IDENTIFYING FUNCTIONS,
DOMAINS, AND RANGES
Decide whether each relation defines a function and
give the domain and range.
Example 5
b. y  2x  1
Solution
2x  1  0
2x  1
1
x
2
Solve the inequality.
Add 1.
Divide by 2.
1 
Domain is  ,   .
2 
2.3 - 29
IDENTIFYING FUNCTIONS,
DOMAINS, AND RANGES
Decide whether each relation defines a function and
give the domain and range.
Example 5
b. y  2x  1
Solution
2x  1  0
2x  1
1
x
2
Solve the inequality.
Add 1.
Divide by 2.
Because the radical is a non-negative number, as x
takes values greater than or equal to ½ , the range
is y ≥ 0 or 0,   .
2.3 - 30
IDENTIFYING FUNCTIONS,
DOMAINS, AND RANGES
Decide whether each relation defines a function and
give the domain and range.
Example 5
2
y
x
c.
Solution Ordered pairs (16, 4) and (16, –4) both
satisfy the equation. Since one value of x, 16,
corresponds to two values of y, this equation does
not define a function.
The domain is 0,   .
Any real number can be squared, so the
range of the relation is (  ,  ).
2.3 - 31
IDENTIFYING FUNCTIONS,
DOMAINS, AND RANGES
Decide whether each relation defines a function and
give the domain and range.
Example 5
d. y  x  1
Solution The ordered pairs (1, 0), (1, –1), (1, –2),
and (1, –3) all satisfy the inequality. An inequality
rarely defines a function. Since any number can be
used for x or for y, the domain and range are the
set of real numbers or (  ,  ).
2.3 - 32
IDENTIFYING FUNCTIONS,
DOMAINS, AND RANGES
Decide whether each relation defines a function and
give the domain and range.
Example 5
5
e. y 
x 1
Solution Substituting any value in for x,
subtracting 1 and then dividing it into 5, produces
exactly one value of y for each value in the domain.
This equation defines a function.
2.3 - 33
IDENTIFYING FUNCTIONS,
DOMAINS, AND RANGES
Decide whether each relation defines a function and
give the domain and range.
Example 5
5
e. y 
x 1
Solution Domain includes all real numbers
except those making the denominator 0.
x 1 0
x 1
Add 1.
The domain includes all real numbers except 1
and is written   ,1  1,   .
The range is the interval   ,0    0 ,   .
2.3 - 34
Function Notation
When a function  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 a function notation, to express this and
read (x) as “ of x.” The letter is he name given
to this function. For example, if y = 9x – 5, we can
name the function  and write
f ( x )  9 x  5.
2.3 - 35
Function Notation
Note that (x) is just another name for the
dependent variable y. Fore example, if y = (x) =
9x – 5 and x = 2, then we find y, or (2), by
replacing x with 2.
f (2)  9 2  5  13
The statement “if x = 2, the y = 13” represents the
ordered pair (2, 13) and is abbreviated with the
function notation as
f (2)  13.
2.3 - 36
Function Notation
f (2)  13
Read “ of 2” or “ at 2.” Also,
f (0)  9 0  5  5 and f ( 3)  9( 3)  5  32.
These ideas can be illustrated as follows.
Name of the function
y

Value of the function
Defining expression
f (x)

9x  5
Name of the independent variable
2.3 - 37
Variations of the Definition of
Function
1. A function is a relation in which, for each
distinct value of the first component of
the ordered pairs, there is exactly one
value of the second component.
2. A function is a set of ordered pairs in
which no first component is repeated.
3. A function is a rule or correspondence
that assigns exactly one range value to
each distinct domain value.
2.3 - 38
Caution The symbol (x) does not
indicate “ times x,” but represents the yvalue for the indicated x-value. As just
shown, (2) is the y-value that corresponds
to the x-value 2.
2.3 - 39
Example 6
USING FUNCTION NOTATION
Let (x) = –x2 + 5x – 3 and g(x) = 2x + 3. Find
and simplify.
a. (2)
Solution
( x )   x 2  5 x  3
(2)  22  5 2  3
  4  10  3
3
Replace x with 2.
Apply the exponent;
multiply.
Add and subtract.
Thus, (2) = 3; the ordered pair (2, 3) belongs to .
2.3 - 40
Example 6
USING FUNCTION NOTATION
Let (x) = –x2 + 5x – 3 and g(x) = 2x + 3. Find
and simplify.
b. (q )
Solution
( x )   x 2  5 x  3
(q )  q 2  5q  3
Replace x with q.
2.3 - 41
Example 6
USING FUNCTION NOTATION
Let (x) = x2 + 5x –3 and g(x) = 2x + 3. Find
and simplify.
c. g (a  1)
Solution
g ( x )  2x  3
g (a  1)  2(a  1)  3
 2a  2  3
Replace x with
a + 1.
 2a  5
2.3 - 42
USING FUNCTION NOTATION
Example 7
For each function, find (3).
a. ( x )  3 x  7
Solution
( x )  3 x  7
( x )  3 x  7
(3)  3(3)  7
Replace x with 3.
(3)  2
2.3 - 43
Example 7
USING FUNCTION NOTATION
For each function, find (3).
b.   ( 3,5),(0,3),(3,1),(6, 1)
Solution For  = {( – 3, 5), (0, 3), (3, 1), (6, – 9)},
we want (3), the y-value of the ordered pair
where x = 3. As indicated by the ordered pair
(3, 1), when x = 3, y = 1,so(3) = 1.
2.3 - 44
Example 7
USING FUNCTION NOTATION
For each function, find (3).
c.
Domain
Range
–2
3
10
6
5
2
Solution
In the mapping, the domain element 3 is
paired with 5 in the range, so (3) = 5.
2.3 - 45
Example 7
USING FUNCION NOTATION
For each function, find (3).
y  ( x )
d.
Solution
Start at 3 on the x-axis
and move up to the
graph. Then, moving
horizontally to the yaxis gives 4 for the
corresponding y-value.
Thus (3) = 4.
4
2
0
2
3 4
2.3 - 46
Finding an Expression for (x)
Consider an equation involving x
and y. Assume that y can be
expressed as a function  of x.
To find an expression for (x):
1. Solve the equation for y.
2. Replace y with (x).
2.3 - 47
Example 8
WRITING EQUATIONS USING
FUNCTION NOTATION
Assume that y is a function of x. Rewrite the
function using notation.
a. y  x 2  1
2
y  x 1
Solution
( x )  x 2  1
Let y = (x)
Now find (–2) and (a).
( 2)  ( 2)  1
2
Let x = –2
 4 1
5
2.3 - 48
Example 8
WRITING EQUATIONS USING
FUNCTION NOTATION
Assume that y is a function of x. Rewrite the
function using notation.
a. y  x 2  1
2
y  x 1
Solution
( x )  x 2  1
Let y = (x)
Now find (–2) and (a).
 (a )  a  1
2
Let x = a
2.3 - 49
Example 8
WRITING EQUATIONS USING
FUNCTION NOTATION
Assume that y is a function of x. Rewrite the
function using notation.
b. x  4 y  5
Solution
x  4y  5
Solve for y.
 4y   x  5
x 5
y
4
1
5
( x )  x 
4
4
Multiply by –1;
divide by 4.
ab a b
 
c
c c
2.3 - 50
Example 8
WRITING EQUATIONS USING
FUNCTION NOTATION
Assume that y is a function of x. Rewrite the
function using notation.
b. x  4 y  5
Solution
Now find
(–2) and
(a).
1
5
7
( 2)  ( 2)   
4
4
4
1
5
 (a )  a 
4
4
Let x = –2
Let x = a
2.3 - 51
Increasing, Decreasing, and
Constant Functions
Suppose that a function  is defined over an
interval I. If x1 and x2 are in I,
(a)  increases on I if, whenever x1 < x2,
(x1) < (x2)
(b)  decreases on I if, whenever x1 < x2,
(x1) > (x2)
(c)  is constant on I if, for every x1 and x2,
(x1) = (x2)
2.3 - 52
Example 9
DETERMINING INTERVALS OVER WHICH A
FUNCTION IS INCREASING, DECREASING, OR
CONSTANT
Determine the intervals over which the function
is increasing, decreasing, or constant.
y
6
2
x
–2
1
3
2.3 - 53
DETERMINING INTERVALS OVER WHICH A
FUNCTION IS INCREASING, DECREASING, OR
CONSTANT
Example 9
Determine the intervals over which the function
is increasing,
decreasing, or constant.
y
Solution
6
2
–2
1
3
On the interval (–, 1), the
y-values are decreasing;
on the interval [1,3], the yvalues are increasing; on
x
the interval [3, ), the yvalues are constant (and
equal to 6).
2.3 - 54
DETERMINING INTERVALS OVER WHICH A
FUNCTION IS INCREASING, DECREASING, OR
CONSTANT
Example 9
Determine the intervals over which the function
is increasing,
decreasing, or constant.
y
Solution
6
2
–2
1
3
Therefore, the function is
decreasing on (–, 1),
x increasing on [1,3], and
constant on [3, ).
2.3 - 55
Example 10
INTERPRETING A GRAPH
Swimming Pool
Water Level
4000
Gallons
This graph shows the
relationship between
the number of gallons,
g(t), of water in a small
swimming pool and
time in hours, t.
Answer the following
questions using the
graph information.
3000
2000
1000
25
50
75 100
Hours
2.3 - 56
Example 10
INTERPRETING A GRAPH
Swimming Pool
Water Level
4000
Gallons
a. What is the
maximum number
of gallons of water
in the pool?
When is the
maximum water
level first
reached?
3000
2000
1000
25
50
75 100
Hours
Solution The max range value is 3000 and the
max number of gallons, 3000, is first reached at
t = 25 hr.
2.3 - 57
Example 10
INTERPRETING A GRAPH
Swimming Pool
Water Level
b. For how long is
the water level
increasing?
Decreasing?
Constant?
Gallons
4000
3000
2000
1000
25
50
75 100
Hours
Solution The water level is increasing for 25 – 0 = 25
hr and is decreasing for 75 – 50 = 25 hr. It is constant
for (50 – 25) + (100 – 75) = 25 + 25 = 50 hr.
2.3 - 58
Example 10
INTERPRETING A GRAPH
Swimming Pool
Water Level
c. How many gallons
of water are in the
pool after 90 hr?
Gallons
4000
3000
2000
1000
25
50
75 100
Hours
Solution When t = 90, y = g (90) = 2000. There are
2000 gal after 90 hr.
2.3 - 59
Example 10
INTERPRETING A GRAPH
Swimming Pool
Water Level
d. Describe a series
of events that
could account for
the water level
changes shown in
the graph.
Gallons
4000
3000
2000
1000
25
50
75 100
Hours
Solution The pool is empty at the beginning, then filled to a
level of 3000 gal during the first 25 hr and the water level then
remains the same. At 50 hr the pool starts to be drained over
25 hr to 2000 gal and remains there for 25 hr.
2.3 - 60