Transcript 1-6 - Mr. Raine`s Algebra 2 Class

1-6
1-6 Relations
Relations and
and Functions
Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
1-6
Relations and Functions
Warm Up
Use the graph for Problems 1–2.
1. List the x-coordinates of the points.
–2, 0, 3, 5
2. List the y-coordinates of the points.
3, 4, 1, 0
Holt Algebra 2
1-6
Relations and Functions
Objectives
Identify the domain and range of relations
and functions.
Determine whether a relation is a function.
Holt Algebra 2
1-6
Relations and Functions
Vocabulary
relation
domain
range
function
Holt Algebra 2
1-6
Relations and Functions
A relation is a pairing of input values with
output values. It can be shown as a set of
ordered pairs (x,y), where x is an input and
y is an output.
The set of input values for a relation is
called the domain, and the set of output
values is called the range.
Holt Algebra 2
1-6
Relations and Functions
Mapping Diagram
Domain
Range
A
2
B
C
Set of Ordered Pairs
{(2, A), (2, B), (2, C)}
(x, y)
Holt Algebra 2
(input, output)
(domain, range)
1-6
Relations and Functions
Example 1: Identifying Domain and Range
Give the domain and range for this relation:
{(100,5), (120,5), (140,6), (160,6), (180,12)}.
List the set of ordered pairs:
{(100, 5), (120, 5), (140, 6), (160, 6), (180, 12)}
Domain: {100, 120, 140, 160, 180} The set of x-coordinates.
Range: {5, 6, 12}
Holt Algebra 2
The set of y-coordinates.
1-6
Relations and Functions
Check It Out! Example 1
Give the domain and range for the relation
shown in the graph.
List the set of ordered pairs:
{(–2, 2), (–1, 1), (0, 0),
(1, –1), (2, –2), (3, –3)}
Domain: {–2, –1, 0, 1, 2, 3} The set of x-coordinates.
Range: {–3, –2, –1, 0, 1, 2} The set of y-coordinates.
Holt Algebra 2
1-6
Relations and Functions
Suppose you are told that a person entered
a word into a text message using the
numbers 6, 2, 8, and 4 on a cell phone. It
would be difficult to determine the word
without seeing it because each number can
be used to enter three different letters.
Holt Algebra 2
1-6
Relations and Functions
Number
{Number, Letter}
{(6, M), (6, N), (6, O)}
{(2, A), (2, B), (2, C)}
{(8, T), (8, U), (8, V)}
{(4, G), (4, H), (4, I)}
Holt Algebra 2
The numbers 6, 2, 8,
and 4 each appear as
the first coordinate of
three different ordered
pairs.
1-6
Relations and Functions
However, if you are told to enter the word MATH
into a text message, you can easily determine
that you use the numbers 6, 2, 8, and 4,
because each letter appears on only one
numbered key.
{(M, 6), (A, 2), (T, 8), (H,4)}
The first coordinate is different
in each ordered pair.
A relation in which the first coordinate is never
repeated is called a function. In a function, there
is only one output for each input, so each element
of the domain is mapped to exactly one element in
the range.
Holt Algebra 2
1-6
Relations and Functions
Although a single input in a function cannot
be mapped to more than one output, two
or more different inputs can be mapped to
the same output.
Holt Algebra 2
1-6
Relations and Functions
Not a function: The
relationship from number to
letter is not a function because
the domain value 2 is mapped to
the range values A, B, and C.
Function: The relationship from
letter to number is a function
because each letter in the domain
is mapped to only one number in
the range.
Holt Algebra 2
1-6
Relations and Functions
Example 2: Determining Whether a Relation is a
Function
Determine whether each relation is a function.
A. from the items in a store to their prices on
a certain date
There is only one price for each different item on
a certain date. The relation from items to price
makes it a function.
B. from types of fruits to their colors
A fruit, such as an apple, from the domain would
be associated with more than one color, such as
red and green. The relation from types of fruits
to their colors is not a function.
Holt Algebra 2
1-6
Relations and Functions
Check It Out! Example 2
Determine whether each relation is a function.
A.
There is only one price for
each shoe size. The relation
from shoe sizes to price
makes is a function.
B. from the number of items in a grocery cart
to the total cost of the items in the cart
The number items in a grocery cart would be
associated with many different total costs of the
items in the cart. The relation of the number of
items in a grocery cart to the total cost of the
items is not a function.
Holt Algebra 2
1-6
Relations and Functions
Every point on a vertical line has the same
x-coordinate, so a vertical line cannot
represent a function. If a vertical line
passes through more than one point on the
graph of a relation, the relation must have
more than one point with the same xcoordinate. Therefore the relation is not a
function.
Holt Algebra 2
1-6
Relations and Functions
Holt Algebra 2
1-6
Relations and Functions
Example 3A: Using the Vertical-Line Test
Use the vertical-line test to determine
whether the relation is a function. If not,
identify two points a vertical line would pass
through.
This is a function. Any vertical
line would pass through only
one point on the graph.
Holt Algebra 2
1-6
Relations and Functions
Example 3B: Using the Vertical-Line Test
Use the vertical-line test to determine
whether the relation is a function. If not,
identify two points a vertical line would pass
through.
This is not a function. A vertical
line at x = 1 would pass through
(1, 1) and (1, –2).
Holt Algebra 2
1-6
Relations and Functions
Check It Out! Example 3a
Use the vertical-line test to determine whether
the relation is a function. If not, identify two
points a vertical line would pass through.
This is a function. Any vertical
line would pass through only
one point on the graph.
Holt Algebra 2
1-6
Relations and Functions
Check It Out! Example 3a
Use the vertical-line test to determine whether
the relation is a function. If not, identify two
points a vertical line would pass through.
This is not a function. A vertical
line at x = 1 would pass
through (1, 2) and (1, –2).
Holt Algebra 2
1-6
Relations and Functions
Lesson Quiz: Part I
1. Give the domain and range for this relation:
{(10, 5), (20, 5), (30, 5), (60, 100), (90, 100)}.
D: {10, 20, 30, 60, 90)}
R: {5, 100}
Determine whether each relation is a function.
2. from each person in class to the number of pets
he or she has function
3. from city to zip code not a function
Holt Algebra 2
1-6
Relations and Functions
Lesson Quiz: Part II
Use the vertical-line test to determine
whether the relation is a function. If not,
identify two points a vertical line would pass
through.
4.
not a function; possible answer: (3, 2) and (3, –2)
Holt Algebra 2
1-7
1-6
Relations
Function Notationand Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
1-6
Relations and Functions
Warm Up
Evaluate.
1. 5x – 2 when x = 4
18
2. 3x2 + 4x – 1 when x = 5 94
3.
when x = 16
48
4. 2 – t2 when
5. Give the domain and range for this
relation: {(1, 1), (–1, 1), (2, 4), (–2, 4),
(–3, 9), (3, 9)}.
D: {–3, –2, –1, 1, 2, 3}
Holt Algebra 2
R: {1, 4, 9}
1-6
Relations and Functions
Objectives
Write functions using function notation.
Evaluate and graph functions.
Holt Algebra 2
1-6
Relations and Functions
Vocabulary
function notation
dependent variable
independent variable
Holt Algebra 2
1-6
Relations and Functions
Some sets of ordered pairs can be described by
using an equation. When the set of ordered
pairs described by an equation satisfies the
definition of a function, the equation can be
written in function notation.
Holt Algebra 2
1-6
Relations and Functions
Output value
Input value
ƒ(x) = 5x + 3
ƒ of x equals 5 times x plus 3.
Holt Algebra 2
Output value
Input value
ƒ(1) = 5(1) + 3
ƒ of 1 equals 5 times 1 plus 3.
1-6
Relations and Functions
The function described by ƒ(x) = 5x + 3 is the same
as the function described by y = 5x + 3. And both of
these functions are the same as the set of ordered
pairs (x, 5x+ 3).
y = 5x + 3
(x, y)
ƒ(x) = 5x + 3
(x, ƒ(x))
(x, 5x + 3)
Notice that y = ƒ(x)
(x, 5x + 3) for each x.
The graph of a function is a picture of the
function’s ordered pairs.
Holt Algebra 2
1-6
Relations and Functions
Caution
f(x) is not “f times x” or “f multiplied by x.” f(x)
means “the value of f at x.” So f(1) represents
the value of f at x =1
Holt Algebra 2
1-6
Relations and Functions
Example 1A: Evaluating Functions
For each function, evaluate ƒ(0), ƒ
ƒ(–2).
ƒ(x) = 8 + 4x
Substitute each value for x and evaluate.
ƒ(0) = 8 + 4(0) = 8
ƒ
=8+4
= 10
ƒ(–2) = 8 + 4(–2) = 0
Holt Algebra 2
, and
1-6
Relations and Functions
Example 1B: Evaluating Functions
For each function, evaluate ƒ(0), ƒ
ƒ(–2).
Use the graph to find the
corresponding y-value for
each x-value.
ƒ(0) = 3
ƒ
=0
ƒ(–2) = 4
Holt Algebra 2
, and
1-6
Relations and Functions
Check It Out! Example 1a
For each function, evaluate ƒ(0), ƒ
ƒ(–2).
ƒ(x) = x2 – 4x
Holt Algebra 2
, and
1-6
Relations and Functions
Check It Out! Example 1b
For each function, evaluate ƒ(0), ƒ
ƒ(–2).
ƒ(x) = –2x + 1
Holt Algebra 2
, and
1-6
Relations and Functions
In the notation ƒ(x), ƒ is the name of the function.
The output ƒ(x) of a function is called the
dependent variable because it depends on the
input value of the function. The input x is called the
independent variable. When a function is
graphed, the independent variable is graphed on
the horizontal axis and the dependent variable is
graphed on the vertical axis.
Holt Algebra 2
1-6
Relations and Functions
Holt Algebra 2
1-6
Relations and Functions
Example 2A: Graphing Functions
Graph the function.
{(0, 4), (1, 5), (2, 6), (3, 7), (4, 8)}
Graph the points.
Do not connect the
points because the
values between the
given points have
not been defined.
Holt Algebra 2
1-6
Relations and Functions
Reading Math
A function whose graph is made up of
unconnected points is called a discrete function.
Holt Algebra 2
1-6
Relations and Functions
Example 2B: Graphing Functions
Graph the function f(x) = 3x – 1.
Make a table.
Graph the points.
x
3x – 1
f(x)
–1
3(– 1) – 1
–4
0
3(0) – 1
–1
1
3(1) – 1
2
Connect the points with a line because
the function is defined for all real numbers.
Holt Algebra 2
1-6
Relations and Functions
Check It Out! Example 2a
Graph the function.
3 5 7 9
2 6 10
Graph the points.
Do not connect the
points because the
values between the
given points have not
been defined.
Holt Algebra 2
1-6
Relations and Functions
Check It Out! Example 2b
Graph the function f(x) = 2x + 1.
Graph the points.
Make a table.
x
2x + 1
f(x)
–1
2(– 1) + 1
–1
0
2(0) + 1
1
1
2(1) + 1
3
Connect the points with a line because
the function is defined for all real numbers.
Holt Algebra 2
1-6
Relations and Functions
The algebraic expression used to define
a function is called the function rule. The
function described by f(x) = 5x + 3 is
defined by the function rule 5x + 3. To
write a function rule, first identify the
independent and dependent variables.
Holt Algebra 2
1-6
Relations and Functions
Example 3A: Entertainment Application
A carnival charges a $5 entrance fee and $2
per ride.
Write a function to represent the total cost
after taking a certain number of rides.
Let r be the number of rides and let C be the total cost
in dollars. The entrance fee is constant.
First, identify the independent and dependent variables.
Cost depends on the entrance fee plus the number of rides taken
Dependent variable
Independent variable
Cost = entrance fee + number of rides taken
C(r) = 5 + 2r
Holt Algebra 2
Replace the words with expressions.
1-6
Relations and Functions
Example 3B: Entertainment Application
A carnival charges a $5 entrance fee and $2
per ride.
What is the value of the function for an input
of 12, and what does it represent?
C(12) = 5 + 2(12)
Substitute 12 for r and simplify.
C(12) = 29
The value of the function for an input of 12 is 29. This
means that it costs $29 to enter the carnival and take 12
rides.
Holt Algebra 2
1-6
Relations and Functions
Check It Out! Example 3a
A local photo shop will develop and print the
photos from a disposable camera for $0.27 per
print.
Write a function to represent the cost of photo
processing.
Let x be the number of photos and let f be the total cost
of the photo processing in dollars.
First, identify the independent and dependent variables.
Cost depends on the number of photos processed
Dependent variable
Independent variable
Cost = 0.27  number of photos processed
f(x) = 0.27x
Holt Algebra 2
Replace the words with expressions.
1-6
Relations and Functions
Check It Out! Example 3b
A local photo shop will develop and print the
photos from a disposable camera for $0.27 per
print.
What is the value of the function for an input
of 24, and what does it represent?
f(24) = 0.27(24)
Substitute 24 of x and simplify.
= 6.48
The value of the function for an input of 24 is 6.48.
This means that it costs $6.48 to develop 24 photos.
Holt Algebra 2
1-6
Relations and Functions
Lesson Quiz: Part I
For each function, evaluate
1. f(x) = 9 – 6x 9; 6; 21
2.
4; 6; 0
3. Graph f(x)= 4x + 2.
Holt Algebra 2
1-6
Relations and Functions
Lesson Quiz: Part II
4. A painter charges $200 plus $25 per can of paint
used.
a. Write a function to represent the total charge
for a certain number of cans of paint.
t(c) = 200 + 25c
b. What is the value of the function for an input
of 4, and what does it represent?
300; total charge in dollars if 4 cans of
paint are used.
Holt Algebra 2