Transcript 2.1: Represent Relations and Functions

Represent Relations and Functions
1.
2.
3.
4.
Objectives:
To find the domain and range of a relation or
function
To determine if a relation is a function
To classify and evaluate functions
To distinguish between discrete and continuous
functions
Vocabulary
Relation
Function
Input
Output
Domain
Range
Independent
Variable
Dependent Variable
Objective 1
You will be
able to find
the domain
and range
of a
relation (or
function)
Relation
A mathematical relation is the pairing up
(mapping) of inputs and outputs.
What’s the domain and range of each relation?
Relations
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
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
Exercise 1
Consider the relation given by the ordered
pairs (3, 2), (-1, 0), (2, -1), (-2, 1), (0, 3).
2. Represent the relation as a graph and as
a mapping diagram
Objective 2
You will be
able to tell if
a relation is
a function
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 bread.
Calvin and Hobbes!
You can’t input
bread and
expect a waffle!
What comes out
of a toaster?
It depends on
what you put in.
What’s Your Function?
A function is a
relation in which
each input has
exactly one output.
• A function is a
dependent relation
• Output depends on
the input
Relations
Functions
What’s Your Function?
A function is a
relation in which
each input has
exactly one output.
• Each output does
not necessarily
have only one input
Relations
Functions
How Many
Girlfriends?
If you think of the
input as a boy and
the output as a girl,
then a function
occurs when each
boy has only one
girlfriend.
Otherwise the boy
gets in BIG trouble.
Functional
Relation
Non-Functional
Relation
Another
Functional Relation
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
Objective 1
You will be
able to find
the domain
and range
of a
relation (or
function)
-Revisited-
Domain and Range: Graphs
Domain: All 𝑥values (L→R)
ℝ
𝑥: 𝑥 ∈ ℝ
−∞, ∞
𝑥: −∞ < 𝑥 < ∞
Domain: All real numbers
Domain and Range: Graphs
Range: All 𝑦values (B→T)
𝑦 ≥ −4
−4, ∞)
𝑦: 𝑦 ≥ −4
Range: Greater than
or equal to -4
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
Easier to ask what you can’t get for 𝑦
Range: What you can get out for 𝑦 using the
domain
Exercise 6
Determine the domain of each function.
1. 𝑦 = 𝑥 2 + 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:
The
denominator
of any
fractions
can’t be
zero
Set denominator ≠ 0 and solve
Set radicand ≥ 0 and solve
Square
roots can’t
be negative
Objective 3
You will be able to
classify and evaluate
functions
Dependent Quantities
Functions can also be thought of as dependent
relations. In a function, the value of the output
depends on the value of the input.
Independent Quantity
Input values
Dependent Quantity
Output values
𝑥-values
𝑦-values
Domain
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 ).
Read “𝑓 of 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 8e
Evaluate each function when 𝑥 = −3.
1. 𝑓 𝑥 = −2𝑥 3 + 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
Exercise 8c
Classify each of the following functions as linear or
nonlinear.
1. 𝑓 𝑥 = −2𝑥 3 + 5
2. 𝑔 𝑥 = 12 − 8𝑥
Exercise 9
For 𝑓 𝑥 = 3𝑥 2 − 5𝑥 + 7, find 𝑓(𝑥 + ℎ).
Objective 4
You will be able to tell the difference
between continuous
and discrete functions
Analog vs. Digital
Analog: A signal
created by some
physical process
 Sound,
temperature, etc.
 Contains an infinite
amount of data
Digital: A numerical
representation of
an analog signal
created by samples
 Not continuous =
set of points
 Contains 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:
A function whose graph
consists of an
unbroken curve
Discrete Function:
A function whose graph
consists of a set of
discontinuous points
Exercise 10
Determine whether each situation describes a
continuous or a discrete function. Then state a
realistic domain.
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.
Exercise 10
Determine whether each situation describes a
continuous or a discrete function. Then state a
realistic domain.
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.
Represent Relations and Functions
1.
2.
3.
4.
Objectives:
To find the domain and
range of a relation or
function
To determine if a
relation is a function
To classify and
evaluate functions
To distinguish between
discrete and
continuous functions