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Transcript
Algebra 2
Relations and Functions
Lesson 2-1 Part 1
Goals
Goal
• To graph relations.
• To identify functions.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced
problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Essential Question
Big Idea: Function
• What are relations and when is a relation a
function?
Vocabulary
•
•
•
•
•
Relation
Domain
Range
Function
Vertical Line Test
Definitions:
• 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.
• Domain – the set of input values for a
relation. The x-coordinate of an ordered
pair.
• Range – the set of output values for a
relation. The y-coordinate of an ordered
pair.
Four Ways to Represent a Relation
•
•
•
•
Ordered Pairs (input, output)
Visually (mapping diagram)
Numerically (table of values)
Graphical (points or curve on graph)
Relation – Ordered Pairs
• Set of ordered pairs: {(-2, 3), (0, 0), (2, 3), (4, -1)}
– Domain (set of inputs): {-2, 0, 2, 4}
– Range (set of outputs): {3, 0, 3, -1}
• Set of ordered pairs: {(-1, 1), (-1, -1), (0, 3), (2, 4)}
– Domain (set of inputs): {-1, 0, 2}
– Range (set of outputs): {1, -1, 3, 4}
• Set of ordered pairs: {(-2, 6), (-1, 6), (0, 6), (1, 6),
(2, 6)}
– Domain (set of inputs): {-2, -1, 0, 1, 2}
– Range (set of outputs): {6}
Your Turn:
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}
The set of y-coordinates.
Relations – Mapping Diagram
Relations can 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 the relation.
Each first component is paired with a second
component.
x-axis values
y-axis values
1
–2
3
2
4
–1
Relations – Mapping Diagram
1.
– Example:
1
-3
7
2.
9
2
-5
4
1
-2
– 1. Relation: {(1,2), (-3,9), (7,-5)}
• Domain: {1, -3, 7}
• Range: {9, 2, -5}
– 2. Relation: {(4,-6), (1,3), (1,0), (-2,7)}
• Domain: {4, 1, -2}
• Range: {3, 0, -6, 7}
3
0
-6
7
Your Turn:
State the relation and give it’s domain and range.
A
2
B
C
Relation:
{(2, A), (2, B), (2, C)}
Domain: {2}
Range: {A, B, C}
Your Turn:
State the relation and give it’s domain and range.
1
5
7
-1
0
2
11
Relation  {(1, 0), (5, 2), (7, 2), (-1, 11)}
Domain  {1, 5, 7, -1}
Range  {0, 2, 11}
Relations – Table of Values
Table of Values:
Relation: {(-1, -4), (0, -1), (1, 2), (2, 5)}
Domain: {-1, 0, 1, 2}
Range: {-4, -1, 2, 5}
Relations – Table of Values
Table of values real world situation.
Relation: {(2001, 20.8), (2002, 20.6),
(2003, 20.8), (2004, 20.9), (2005, 21.6),
(2006, 22.3), (2007, 22.6)}.
Domain: {2001, 2002, 2003, 2004,
2005, 2006, 2007}.
Range: 20.8, 20.6, 20.8, 20.9, 21.6,
22.3, 22.6}.
Your Turn:
State the relation and give it’s domain and range.
Relation: {(1, 12.5), (2, 16), (4, 22),
(6, 24), (9, 25), (12, 26)}.
Domain: {1, 2, 4, 6, 9, 12}.
Range: {12.5, 16, 22, 24, 25, 26}.
Relations – Graphical
Example:
Relation, individual points on a graph.
y
The ordered pairs in the relation:
(– 3, 0), (– 2, 2), (– 1, 4), (0, 1),
(1 – 2), and (2, 1).
4
3
2
1
x
4 3 2 1
2
3
4
1
2
3
4
The domain is the set of all
x-coordinates: {– 3, – 2, – 1, 0, 1, 2}.
The range is the set of all
y-coordinates: {0, 2, 4, 1, – 2, 1}.
Relations – Graphical
Example:
Relation, straight line or curve on a graph.
y
Relation:
4
2
y   x  2.
3
3
2
1
x
4 3 2 1
2
3
4
1
2
3
4
The domain: all real numbers.
The range: all real numbers.
Relations – Graphical
Example:
Relation, straight line or curve on a graph.
Domain: {-5 ≤ x ≤ 9}
Range: {-5 ≤ y ≤ 5}
Your Turn:
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.
Your Turn:
Give the domain and range of each relation.
y
(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 - 20
Your Turn:
Give the domain and range of each relation.
y
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≤x≤4}.
The range is {–6≤x≤6}.
–6
2.3 - 21
Your Turn:
Give the domain and range of each relation.
y
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.
2.3 - 22
Your Turn:
Give the domain and range of each relation.
y
The arrowheads indicate
that the line extends
indefinitely left and right,
as well as upward. The
domain is all real
numbers. Because there
x is at least y-value, –3, the
range includes all
numbers greater than, or
equal to –3 or {y ≥ -3}.
2.3 - 23
Definition:
Function
A function is a relation in which, for each
distinct value of the first component of the
ordered pair (x-value), there is exactly one
value of the second component (y – value).
Example: Function
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.
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)}
The numbers 6, 2, 8,
and 4 each appear as
the first coordinate of
three different ordered
pairs.
Example: Function
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.
Function:
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.
10
1
2
11
3
4
5
6
14
Function:
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.
Example:
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 - 29
Example:
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 - 30
Example:
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 - 31
Example:
Give the domain and range of the relation.
Tell whether the relation defines a function.
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 - 32
Example:
Give the domain and range of the relation.
Tell whether the relation defines a function.
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 - 33
Your Turn:
Determine whether the relation is a function.
x-axis values
y-axis values
–4
–2
1
0
Not a Function
Your Turn:
Determine whether the relation is a function.
There is only one price for
each shoe size. The relation
from shoe sizes to price
makes is a function.
Your Turn:
Determine whether the relation is a function.
{(10, 5), (20, 5), (30, 5), (60, 100), (90, 100)}
Not a Function
Your Turn:
Determine whether the relation is a function.
y
5
4
3
2
1
0
-4
-3
-2
-1
0
1
-1
-2
-3
Function
2
3
4
x
Your Turn:
Determine whether the relation is a function.
Joe
Samantha
Laura
Anna
Julie
Ian
Hilary
Chelsea
Barbara
George
Sue
Humans
Function
Mothers
Your Turn:
Are these functions?
{ (0, 2), (1, 0), (2, 6), (8, 12) }
{ (0, 2), (1, 0), (2, 6), (8, 12), (9, 6) }
{ (3, 2), (1, 0), (2, 6), (8, 12), (3, 5), }
{ (3, 2), (1, 2), (2, 2), (8, 2), (7, 2) }
{ (1, 1), (1, 2), (1, 5), (1, -3), (1, -5) }
Vertical Line Test
Every point on a vertical line has the same xcoordinate, 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 x-coordinate. Therefore the relation is not a
function.
Example:
Use the vertical line test to determine whether each
relation graphed is a function.
y
(1, 2)
(– 1, 1)
This graph
represents a
x function.
(0, – 1)
(4, – 3)
2.3 - 41
Example:
Use the vertical line test to determine whether each
relation graphed is a function.
y
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 a
function.
–6
2.3 - 42
Example:
Use the vertical line test to determine whether each
relation graphed is a function.
y
x
This graph
represents a
function.
2.3 - 43
Example:
Use the vertical line test to determine whether each
relation graphed is a function.
y
x
This graph
represents a
function.
2.3 - 44
Your Turn:
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.
Your Turn:
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).
Your Turn:
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.
Your Turn:
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).
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.
Essential Question - Review
Big Idea: Function
• What are relations and when is a relation a
function?
• A relation is a set of pairs of input and output
values. A function is a relation in which each
input corresponds with exactly one output.
Assignment:
• Section 2-1 part 1, Pg 68 – 69; #1 – 7 all,
8 – 22 even.