Transcript Relations and Functions

Relations and Functions
Section 1-6 and 1.7
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Review

A relation is a pairing of input values with
output values.

A relation may be viewed as ordered pairs,
mapping design, table, equation, or written in
sentences

x-values are inputs, domain, independent
variable, cause

y-values are outputs, range, dependent
variable, effect
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Example 1
{(0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0)}
•What is the domain?
{0, 1, 2, 3, 4, 5}
What is the range?
{-5, -4, -3, -2, -1, 0}
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Example 2
Input
4
–5
0
–2
Output
9
–1
7
•What is the domain?
{4, -5, 0, 9, -1}
•What is the range?
{-2, 7}
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Is a relation a function?
What is a function?
According to the textbook, “a
function is…a relation in which
every input is paired with
exactly one output”
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Is a relation a function?
•Focus on the x-coordinates, when given a relation
If the set of ordered pairs have different x-coordinates,
it IS A function
If the set of ordered pairs have same x-coordinates,
it is NOT a function
•Y-coordinates have no bearing in determining
functions
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Example 1
{(0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0)}
•Is this a function?
•Hint: Look only at the x-coordinates
YES
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Example 2
{(–1, 7), (1, 0), (2, 3), (0, 8), (0, 5), (–2, 1)}
•Is this a function?
•Hint: Look only at the x-coordinates
NO
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Example 5
Which mapping represents a function?
Choice One
3
1
0
–1
2
3
Choice Two
2
–1
3
2
3
–2
0
Choice 1
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Example 6
Which mapping represents a function?
A.
B.
B
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Example 7
Which situation represents a function?
a. The items in a store to their prices on a
certain date
b. Types of fruits to their colors
There is only one price for each
different item on a certain date. The
relation from items to price makes it a
function.
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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.
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Vertical Line Test
•Vertical Line Test: a relation is a function
if a vertical line drawn through its graph,
passes through only one point.
AKA: “The Pencil Test”
Take a pencil and move it from left to right
(–x to x); if it crosses more than one point,
it is not a function
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Vertical Line Test
Would this
graph be a
function?
YES
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Vertical Line Test
Would this
graph be a
function?
NO
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Is the following function discrete or continuous?
What is the Domain? What is the Range?
Discrete



-7, 1, 5, 7, 8, 10



1, 0, -7, 5, 2, 8
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Is the following function discrete or continuous?
What is the Domain? What is the Range?
continuous



8,8




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6,6
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
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Is the following function discrete or continuous?
What is the Domain? What is the Range?
continuous



0,45




10,70
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
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Is the following function discrete or continuous?
What is the Domain? What is the Range?
discrete



-7, -5, -3, -1, 1, 3, 5, 7

2, 3, 4, 5, 7
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Domain and Range in Real Life
The number of shoes in x pairs of shoes can be
expressed by the equation y = 2x.
What subset of the real numbers makes sense
for the domain?
Whole numbers
What would make sense for the range
of the function?
Zero and the even numbers
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Domain and Range in Real Life
The number of shoes in x pairs of shoes can be
expressed by the equation y = 2x.
What is the independent variable?
The # of pairs of shoes.
What is the dependent variable?
The total # of shoes.
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Domain and Range in Real Life
Mr. Landry is driving to his hometown. It takes four hours to
get there. The distance he travels at any time, t, is
represented by the function d = 55t (his average speed is
55mph.
Write an inequality that represents the domain in real life.
0 x  4
Write an inequality that represents the range in real life.
0  y  220
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Domain and Range in Real Life
Mr. Landry is driving to his hometown. It takes four hours to
get there. The distance he travels at any time, t, is
represented by the function d = 55t (his average speed is
55mph.
What is the independent variable?
The time that he drives.
What is the dependent variable?
The total distance traveled.
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Domain and Range in Real Life
Johnny bought at most 10 tickets to a concert for him and
his friends. The cost of each ticket was $12.50.
Complete the table below to list the possible domain and
range.
1
2
3
12.50 25.00 37.50
4
5
6
50 62.50 75
7
8
9
10
87.50 100 112.50 125
What is the independent variable?
The number of tickets bought.
What is the dependent variable?
The total cost of the tickets.
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Domain and Range in Real Life
Pete’s Pizza Parlor charges $5 for a large pizza with no
toppings. They charge an additional $1.50 for each of
their 5 specialty toppings (tax is included in the price).
Jorge went to pick up his order. They said his total bill
was $9.50. Could this be correct? Why or why not?
Yes
One pizza with 3 toppings cost $9.50
Susan went to pick up her order. They said she owed
$10.25. Could this be correct? Why or why not?
No One pizza with 4 toppings cost $11
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Domain and Range in Real Life
Pete’s Pizza Parlor charges $5 for a large pizza with no
toppings. They charge an additional $1.50 for each of
their 5 specialty toppings (tax is included in the price).
What is the independent variable?
The number of toppings
What is the dependent variable?
The cost of the pizza
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Function Notation
f(x) means function of x and is read “f of x.”
f(x) = 2x + 1 is written in function notation.
The notation f(1) means to replace x with 1 resulting in
the function value.
f(1) = 2x + 1
f(1) = 2(1) + 1
f(1) = 3
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Function Notation
Given g(x) = x2 – 3, find g(-2) .
g(-2) = x2 – 3
g(-2) = (-2)2 – 3
g(-2) = 1
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Function Notation
Given f(x) =
2 x2  3 x
a. f(3)
f(3) = 2x2 – 3x
f(3) = 2(3)2 – 3(3)
f(3) = 2(9) - 9
f(3) = 9
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, the following.
b. 3f(x)
3f(x) = 3(2x2 – 3x)
3f(x) = 6x2 – 9x
c. f(3x)
f(3x) = 2x2 – 3x
f(3x) = 2(3x)2 – 3(3x)
f(3x) = 2(9x2) – 3(3x)
f(3x) = 18x2 – 9x
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Function Notation
Given f(x) =
2 x2  3 x
d. f(-x)
f(-x) = 2x2 – 3x
f(-x) = 2(-x)2 – 3(-x)
f(-x) = 2x 2 + 3x
e. -f(x)
, the following.
f. f(x +3)
f(x+3) = 2x2 – 3x
f(x+3) = 2(x+3)2 – 3(x+3)
f(x+3) = 2(x2 + 6x + 9) – 3x – 9
f(x+3) = 2x2 + 12x + 18 – 3x – 9
f(x+3) = 2x2 + 9x + 9
– f(x) = –(2x2 – 3x)
– f(x) = –2x2 + 3x
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For each function, evaluate f(0), f(1.5), f(-4),
f(0) = 3
f(1.5) = 4
f(4) = -1
f(x) = 5 at x = -5
x=1
f(x) = 1 at x = -1
x=3
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For each function, evaluate f(0), f(1.5), f(-4),
f(0) = -5
f(1.5) = 1
f(-4) = 1
f(x) = -5 at x = -5
x=0
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3
-3
positive
negative
X=-3 x=6 x=10
(3,6) or (10,11
 6,11
3, 4
X=0 x=4
X=-5 x=8
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