Download 4.6: Formalizing Relations and Functions

4.6: Formalizing
Relations and
Functions
Objective
Objective
•To determine whether
a relation is a
function.
Objective
•To determine whether
a relation is a
function.
•To find domain and
range using function
notation.
Vocab (paragraph): page
268
• A relation is a pairing of
numbers in one set, called
the domain, with numbers
in another set, called the
range.
Vocab (paragraph): page
268
• A relation is often
represented as a set of
ordered pairs (x, y). In this
case, the domain is the set
of x-values and the range
is the set of y-values.
Essential Understanding
A function is a special
type of relation in
which each value in
the domain is paired
with exactly one value
in the range.
Essential Understanding
In short, there
can’t be 2 y’s for
the same x.
Problem 1 page 268
Problem 1 page 268
•The diagram they
use is completely
optional;
Problem 1 page 268
•The diagram they use
is completely
optional; that being
said, it may be quite
helpful for you to sort
out your information.
Got it on the top of page
269
a. (4.2, 1.5), (5, 2.2), (7,
4.8), (4.2, 0)
Got it on the top of page
269
a. (4.2, 1.5), (5, 2.2), (7,
4.8), (4.2, 0)
Domain Range
4.2
1.5
5
2.2
7
4.8
0
Got it on the top of page
269
a. (4.2, 1.5), (5, 2.2), (7, 4.8),
(4.2, 0)
Domain Range
4.2
1.5
0
5
2.2
7
4.8
Got it on the top of page
269
a. (4.2, 1.5), (5, 2.2), (7, 4.8),
(4.2, 0)
Domain Range
4.2
1.5
0
5
2.2
7
4.8
Since 4.2
goes to 2
different y
values, this
is not a
function.
Got it on the top of page
269
b. (-1, 1), (-2, 2), (4, -4),
(7, -7)
Dom
ain
-1
-2
4
7
Ra
nge
1
2
-4
-7
Since every x
goes to one y,
this is an
example of a
function..
2nd way of determining:
2nd way of determining:
• Called the vertical line test.
2nd way of determining:
• Called the vertical line test.
• Basically, after graphing
the function, if you can
draw a vertical line
through 2 different points
on the graph, it is not a
function.
Example: y =
𝑥
Example:
Keep in mind
that the square
root of a number
can be positive or
negative.
Example:
For example,
the square root
of 4 is both 2
and -2
Keep in mind
that the square
root of a number
can be positive or
negative.
Example:
For example,
the square root
of 4 is both 2
and -2
Keep in mind
that the square
root of a number
can be positive or
negative.
Since x (4) can
be mapped to
both 2 and -2,
this is not a
function
Example:
For example,
the square root
of 4 is both 2
and -2
Keep in mind
that the
square root of
a number can
be positive or
negative.
Since x (4) can
be mapped to
both 2 and -2,
this is not a
function
Validated by
the vertical
line test.
2nd part of formalizing:
2nd part of formalizing:
Notation
2nd part of formalizing:
Basically, we are
replacing the dependent
variable
2nd part of formalizing:
Basically, we are
replacing the dependent
variable (often y)
2nd part of formalizing:
Basically, we are
replacing the dependent
variable (often y) with
the notation f(x).
2nd part of formalizing:
Basically, we are
replacing the dependent
variable (often y) with
the notation f(x).
y = mx + b
2nd part of formalizing:
Basically, we are
replacing the dependent
variable (often y) with
the notation f(x).
f(x) = mx + b
Evaluating functions…
Evaluating functions…
•Given the function f(x),
replace x with the value
assigned and compute
arithmetically.
Problem 3 on the bottom
of page 269
Problem 3 on the bottom
of page 269
• w(x) = 250x
• Represents the words you
can read in 1 minute
Problem 3 on the bottom
of page 269
• w(x) = 250x
• If they ask how many
words you can read in 8
minutes, they’re saying…
• w(x) = 250x
• If they ask how many
words you can read in 8
minutes, they’re saying…
x=8
• w(x) = 250x
• Replace x with 8
x=8
• w(8) = 250(8)
• Replace x with 8
x=8
which is 2000
• w(8) = 250(8)
• Replace x with 8
x=8
Finding the range of a
function given f(x) notation
Finding the range of a
function given f(x) notation
Given all the values of the
domain.
Finding the range of a
function given f(x) notation
Given all the values of the
domain.
(1) Plug in each value of the
domain into the function
expression.
Finding the range of a
function given f(x) notation
Given all the values of the
domain.
(2) Evaluate the expression.
Finding the range of a
function given f(x) notation
Given all the values of the
domain.
(3) List the results of this as
your range.
Example:
• The domain of f(x) = -1.5x +
4 is {1, 2, 3, 4}. What is the
range?
Example:
• The domain of f(x) = -1.5x +
4 is {1, 2, 3, 4}. What is the
range?
Domain
1
2
3
4
F(x)
Range
Example:
• The domain of f(x) = -1.5x +
4 is {1, 2, 3, 4}. What is the
range?
Domain
F(x)
1
-1.5(1)+ 4
2
3
4
Range
Example:
• The domain of f(x) = -1.5x +
4 is {1, 2, 3, 4}. What is the
range?
Domain
F(x)
1
-1.5(1)+ 4
2
3
4
Range
2.5
Example:
• The domain of f(x) = -1.5x +
4 is {1, 2, 3, 4}. What is the
range?
Domain
F(x)
1
-1.5(1)+ 4
2
-1.5(2) + 4
3
-1.5(3) + 4
4
-1.5(4) + 4
Range
2.5
Example:
• The domain of f(x) = -1.5x +
4 is {1, 2, 3, 4}. What is the
range?
Domain
F(x)
1
-1.5(1)+ 4
2
-1.5(2) + 4
3
-1.5(3) + 4
4
-1.5(4) + 4
Range
2.5
1
-0.5
-2
Example:
• Thus, the range is {-2, -0.5,
1, 2.5}
Domain
F(x)
1
-1.5(1)+ 4
2
-1.5(2) + 4
3
-1.5(3) + 4
4
-1.5(4) + 4
Range
2.5
1
-0.5
-2
Quickly do the got it
underneath.
Quickly do the got it
underneath.
{-8, 0, 8, 16}
Problem 5 to finish…