# 2.1 Represent Relations and Functions by chenmeixiu

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```									2.1: Represent Relations and Functions

Objectives:
1.   To determine if a relation is a function
2.   To find the domain and range of a relation
or function
3.   To classify and evaluate functions
4.   To distinguish between discrete and
continuous functions
Vocabulary
As a group, define      Relation      Function
each of these
Give an example of
each word and          Domain        Range
leave a bit of space
Independent   Dependent
revisions.
Relation
A mathematical relation is the pairing up
(mapping) of inputs and outputs.
Relation
A mathematical relation is the pairing up
(mapping) of inputs and outputs.

• Domain: the set of all input values
• Range: the set of all output values
Relation
A mathematical relation is the pairing up
(mapping) of inputs and outputs.

What’s the domain and range of each relation?
Exercise 1
Consider the relation given by the ordered
pairs (3, 2), (-1, 0), (2, -1), (-2, 1), (0, 3).
1. Identify the domain and range
2. Represent the relation as a graph and as
a mapping diagram
Calvin and Hobbes!

A toaster is an example of a function. You
put in bread, the toaster performs a
toasting function, and out pops toasted
Calvin and Hobbes!

“What comes out of a toaster?”
“It depends on what you put in.”
– You can’t input bread and expect a waffle!
A function is a relation
in which each input
Relations
has exactly one
output.
• A function is a          Functions
dependent relation
• Output depends on
the input
A function is a relation
in which each input
Relations
has exactly one
output.
• Each output does not     Functions
necessarily have only
one input
How Many Girlfriends?
If you think of the
inputs as boys and
the output as girls,
then a function
occurs when each
boy has only one
girlfriend. Otherwise
the boy gets in BIG
trouble.
What’s a Function Look Like?
What’s a Function Look Like?
What’s a Function Look Like?
What’s a Function Look Like?
Exercise 2a
Tell whether or not each table represents a
function. Give the domain and range of
each relationship.
Exercise 2b
The size of a set is called its cardinality.
What must be true about the cardinalities
of the domain and range of any function?
Exercise 3
Which sets of ordered pairs represent
functions?
1. {(1, 2), (2, 3), (3, 4), (3, 5), (5, 6)}
2. {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}
3. {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1)}
4. {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}
Exercise 4
Which of the
following
graphs
represent
functions?
What is an
easy way to
tell that each
input has only
one output?
Vertical Line Test
A relation is a function iff no vertical line
intersects the graph of the relation at more
than one point
If it does,
then an
input has
more than
one
output.
Function       Not a Function
Domain and Range: Graphs
• Domain: All x-
values (L → R)
Range:
– {x: -∞ < x < ∞}
Greater
than or
equal
• Range: All y-                                   to -4
values (D ↑ U)
– {y: y ≥ -4}       Domain: All real numbers
Exercise 5
Determine the domain and range of each
function.
Domain and Range: Equations
• Domain: What you are allowed to plug in
for x.
– Easier to ask what you can’t plug in for x.

• Range: What you can get out for y using
the domain.
– Easier to ask what you can’t get for y.
Exercise 6
Determine the domain of each function.
1. y = x2 + 2

2. y  x  2

1
3. y 
x2
Protip: Domains of Equations
When you have to find the domain of a
function given its equation there’s really
only two limiting factors:
1. The denominator of any fractions can’t be
zero
– Set denominators ≠ zero and solve
2. Square roots can’t be negative
– Set square roots ≥ zero and solve
Dependent Quantities
Functions can also be thought of as
dependent relationships. In a function, the
value of the output depends on the value
of the input.
• Independent quantity: Input values, x-
values, domain
• Dependent quantity: Output value, which
depends on the input value, y-values,
range
Exercise 7
The number of pretzels, p, that can be
packaged in a box with a volume of V cubic
units is given by the equation p = 45V + 10.
In this relationship, which is the dependent
variable?
Function Notation
In an equation, the dependent variable is
usually represented as f (x).
–   f = name of function; x = independent variable
–   Takes place of y
–   f (x) does NOT mean multiplication!
–   f (3) means “the function evaluated at 3”
where you plug 3 in for x.
Exercise 8
Evaluate each function when x = −3.
1. f (x) = −2x3 + 5

2. g (x) = 12 – 8x
Flavors of Functions
Functions come in a variety of flavors. You
will need to be able to distinguish a linear
from a nonlinear function.
Linear Function      Nonlinear Function
f (x) = 3x             f (x) = x2 – 2x + 5
g (x) = ½ x – 5        g (x) = 1/x
h (x) = 15 – 5x        h (x) = |x| + 2
Analog vs. Digital
• Analog: A signal created by some physical
process
– Sound, temperature, etc.
– Contain an infinite amount of data
• Digital: A numerical representation of an
analog signal created by samples
– Not continuous = set of points
– Contain a finite amount of data
Digital Signal Processing
Original Analog Signal

Digital Samples
Digital Signal Processing
Digital signal processing is about converting
an analog signal into digital information,
doing something to it, and usually
converting it back into an analog signal.
Continuous vs. Discrete
• Continuous Function      • Discrete Function
– A function whose         – A function whose
graph consists of an       graph consists of a set
unbroken curve             of discontinuous points
Exercise 9
Determine whether each situation describes
a continuous or a discrete function.
1. The cheerleaders are selling candy bars for
\$1 each to pay for new pom-poms. The
function f (x) gives the amount of money
collected after selling x bars.
2. Kenny determined that his shower head
releases 1.9 gallons of water per minute.
The function V(x) gives the volume of water
released after x minutes.
Assignment
• P. 76-79: 1-3, 9-21
odd, 24, 34-38, 41,
45, 46, 51, 52
• P. 81: 1-9 odd
• Working with
Domain and Range
Worksheet: 1-3, 6-
11, 13, 14, 15, 16,
18, 20, 21

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