Objectives:
1. To determine if a
relation is a function
2. To find the domain
and range of a
function
3. To evaluate
functions
•
•
•
•
Assignment:
P. 48-9: 1-37 odd
P. 50-1: 57-73 odd,
87, 88, 92
P. 61-2: 1-5, 8, 9-14
Read: None
Relation
Function
Input
Output
Domain
Range
Set-Builder Notation
Interval Notation
Function Notation
A mathematical relation is the pairing up
(mapping) of inputs and outputs.
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
A toaster is an example of a function. You put in
bread, the toaster performs a toasting
function, and out pops toasted bread.
“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
has exactly one
output.
• A function is a
dependent relation
• Output depends on
the input
Relations
Functions
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
Tell whether or not each table represents a
function. Give the domain and range of each
relationship.
The size of a set is called its cardinality. What
must be true about the cardinalities of the
domain and range of any function?
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)}
Which of the
following
graphs
represent
functions?
What is an
easy way to
tell that
each input
has only one
output?
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
To determine if an equation represents a
function, try solving the thing for y.
• Make sure that there is only one value of y for
every value of x.
Determine whether each equation represents y
as a function of x.
1. x2 +2y = 4
2. (x + 3)2 + (y – 5)2 = 36
You will be
able to find
the domain
and range of
a function.
Since the domain or range of a function is often
an infinite set of values, it is often convenient
to represent your answers in set-builder
notation.
Examples:
• {x | x < -2} reads “the set of all x such that x is
less than negative 2”.
Since the domain or range of a function is often
an infinite set of values, it is often convenient
to represent your answers in set-builder
notation.
Examples:
• {x : x < -2} reads “the set of all x such that x is
less than negative 2”.
Another way to describe an infinite set of
numbers is with interval notation.
• Parenthesis indicate that first or last number
is not in the set:
𝑥<2
−∞, 2
• Neither the negative infinity
or the 2 are included in the
interval
• Always write the smaller
number, bigger number
Another way to describe an infinite set of
numbers is with interval notation.
• Brackets indicate that first or last number is in
the set:
𝑥≤2
−∞, 2
• Infinity (positive or negative)
never gets a bracket
• Always write the smaller
number, bigger number
Domain: All xvalues (L → R)
– {x: -∞ < x < ∞}
Range: All y-values
(D ↑ U)
– {y: y ≥ -4}
Domain: All real numbers
Range:
Greater
than or
equal
to -4
Determine the domain and range of each
function.
Domain: What you are allowed to plug in for x.
– Easier to ask what you can’t plug in for x.
– Limited by division by zero or negative even roots
– Can be explicit or implied
Range: What you can get out for y using the
domain.
– Easier to ask what you can’t get for y.
Determine the domain of each function.
1. y = x2 + 2
1
2. y  2
x 9
Determine the domain of each function.
1. y  x  2
2
y

x
2
2.
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. A square root can’t be negative
– Set square roots ≥ zero and solve
You will be able
to evaluate
functions.
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
Dependent Quantity
Input values
Output values
x-values
Domain values
Function
y-values
Range values
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?
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: y = f (x)
f (x) does NOT mean multiplication!
f (3) means “the function evaluated at 3” where
you plug 3 in for x.
Evaluate each function when x = −3.
1. f (x) = −2x3 + 5
2. g (x) = 12 – 8x
Let g(x) = −x2 + 4x + 1. Find each function value.
1. g(2)
2. g(t)
3. g(t + 2)
Objectives:
1. To determine if a
relation is a function
2. To find the domain
and range of a
function
3. To evaluate
functions
Assignment:
• P. 48-9: 1-37 odd
• P. 50-1: 57-73 odd,
87, 88, 92
• 61-2: 1-5, 8, 9-14