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Lecture 5.2
Contemporary Mathematics
Instruction: Functions
A BIG idea in mathematics is the idea of a "function." A function is a special type of
relation. Since relations are subsets of Cartesian products, functions are special subsets of
Cartesian products whose elements are ordered pairs such that each first coordinate is paired with
one and only one second coordinate.
Consider the sets X and Y such that X = Y = \ . The Cartesian product of X and Y is \ 2 .
The set X is called the domain and set Y is called the co-domain. A function f is a subset of \ 2
such that each element in the domain X is paired with one and only one element in the co-domain
Y. The set R comprised of the second coordinates in f is called the range, which is a subset of the
co-domain.
A function f is a proper subset of X × Y (where X and Y are non-empty sets) with the
property that each element x in X is paired with one and only one element y in Y. The set
X is called the domain. The set Y is called the co-domain. The set R of elements y such
that there is an element (x,y) in f is called the range and is a subset of the co-domain. A
function f is denoted f : X 6 R , which reads "a function from X to R."
The following set of ordered pairs represents a function since no two ordered pairs have
the same first coordinate and different second coordinates.
F = {( −2, 4 ) , ( −1,1) , ( 0, 0 ) , (1,1) , ( 2,4 )}
The next set of ordered pairs, however, does not represent a functional set since it contains
ordered pairs with the same first coordinate and different second coordinates.
V = {( 4, −2 ) , (1, − 1) , ( 0, 0 ) , (1,1) , ( 4,2 )}
Note that V contains the ordered pairs ( 4, −2 ) and ( 4, 2 ) . Since the first coordinate of 4 has two
distinct corresponding second coordinates, 2 and –2, relation V is not a function.
Consider the sets A = {4, 5, 6} and B = {7, 8} . The Cartesian product of A and B is
given below.
A × B = {( 4, 7 ) , ( 4,8 ) , ( 5, 7 ) , ( 5,8 ) , ( 6, 7 ) , ( 6,8 )}
Let R ⊂ B such that R = {7}. The set f below is a function, f : A 6 R .
f = {( 4, 7 ) , ( 5, 7 ) , ( 6, 7 )}
For f = {( 4, 7 ) , ( 5, 7 ) , ( 6, 7 )} , set A is the domain, set B is the co-domain, and set R = {7} is
the range. This type of function is called a constant function because the second coordinate is
constantly the same, i.e., always seven.
Functions do not have to involve ordered pairs of numbers. Consider set K and C below.
Lecture 5.2
K = {Andy, Billy, Debbie, Kim}
C = {licorice stick, bubble gum, chocolate bar, gumball, lollipop}
Imagine that set K represents four sibling children while C represents the kinds of candy sold at a
local store. Imagine also that the mother of the children treats her daughters with a lollipop and
her sons with a chocolate bar during a shopping trip to the store. The set below represents a
function f : K 6 R where the range R = {lollipop, chocolate bar} .
f = {( Andy, chocolate bar ) , ( Billy, chocolate bar ) , ( Debbie, lollipop ) , ( Kim, lollipop )}
In this example, set K is the domain, and set C is the co-domain.
The previous example helps convey the idea of a function, but we will mostly be
interested in functions that are sets of ordered pairs of real numbers. If we restrict our discussion
to relations and functions that are sets of ordered pairs of real numbers that represent points on
the Cartesian plane, we can employ a simple test called the vertical line test to see if any relation
is a function. In a function, each value of the domain corresponds to one and only one value
from the co-domain, so any vertical line drawn through the graph of a function intersects the
function only once.
Let X = Y = \ . Let X × Y = \ 2 . Let the horizontal axis of the Cartesian Plane
represent X and the vertical axis represent Y. Let H be a relation graphed on the
Cartesian plane. If any vertical line intersects the graph of H at more than one
point, then H is not a function and is said to fail the vertical line test. If any
vertical line intersects the graph of H at no more than one point, then H is a
function and is said to pass the vertical line test.
Figure 1 shows a relation that passes the vertical line test while Figure 2 shows a relation that
fails the vertical line test.
y
y
x
Figure 1
x
Figure 2
The concept of a function is not difficult. Despite its simplicity (or because of its
simplicity), the concept of a function lies at the center of an aesthetic debate in mathematics.
Some mathematicians find the definition above aesthetically displeasing since the definition
focuses on what a function is rather than what a function does. A function can be viewed as a
rule that maps or transforms each element from the domain to one element in a subset of the codomain called the range.
Lecture 5.2
Let x ∈ X , y ∈ Y , and R ⊆ Y . A function f is a rule that maps or transforms elements
from X called the domain to one and only one element from Y called the co-domain. If
( x, y ) ∈ f , then y is the image of x under f, and we say f maps or transforms x to y.
Moreover, the set R with elements y such that there is a pair ( x, y ) in f is called the
range of f, and we write f : X 6 R to indicate "a function from X to R" where range R
is a subset of co-domain Y.
This alternative definition allows certain equations to serve as functions. Consider two sets: X,
the domain, and Y, the co-domain. Let X = Y = \ . The equation y = x 2 represents a function
f : x 6 x 2 where the range R = { y : y ≥ 0, y ∈ \} . The restriction y ≥ 0 is a result of the fact
that y = x 2 and the square of any real number is non-negative. The following lecture will
elaborate further on the domain, co-domain, and range of functions.
.
An Informal Discussion . . .
Let's think of a paperboy who gets paid per subscriber. Let's assume that the paperboy delivers
papers to x number of subscribers. If the newspaper pays the paperboy $4.50 per month per
subscriber, we can write a monthly income function for the paperboy:
f : subscribers 6 $
y = $4.50x
We will think of the domain elements as inputs in the function and the range elements as outputs.
The "rule" says multiply the input by 4.5 to get the output. The input is the number of
subscribers. The monthly income is the output of the function. For example, if the paperboy has
100 subscribers, he earns $450.00 per month, i.e., if x = 100, y = $450.00.
A realistic domain for this function would be D = { x : 0 ≤ x ≤ 500, x ∈ ]} . A realistic range for
the function would be R = { y : $4.50 ≤ x ≤ $2, 250.00, y equals a multiple of $4.50} .
Application Exercise 5.2
Problems
#1
For a science project, Gregor stands in his driveway at the top of each hour after sunrise until
sunset and records the length of his shadow. Use descriptive notation to identify the domain and
range of this function.
#2
Consider an experiment measuring the intelligence of a bird using a food pellet dispenser. If the
bird pecks at a green button, one food pellet drops. If the bird pecks at a blue button, two food
pellets drop. If the bird pecks at a red button, three food pellets drop. Describe this situation
using function notation.
#3
Consider the formula that converts Fahrenheit temperature readings to Celsius and fill in the blank
below.
5
C = ( F − 32 )
9
The function f : F ° → C ° maps a Fahrenheit reading of 41° to ______________________.
#4
Let X = Y = \ . Determine if the following subsets of X × Y are functions or relations.
A = {( −2, 4 ) ,
( −1,1) , ( 0, 0 ) , (1,1) , ( 2, 4 )}
B = {( 0, 0 ) , (1,1) , ( 4, 2 ) , ( 9, 3) , (16, 4 )}
C = {( 0, 0 ) , (1,1) , (1, −1) , ( 2, 4 ) , ( 2, −4 )}
#5
Let X = Y = \ . Determine if the following graphs of H, G, J, and K (all subsets of X × Y ) are
functions or relations.
y
y
y
y
G
H
x
K
x
x
x
J
#1 domain = {the hours of the day} , range = {Gregor's shadow length measurements}
#2
#3
#4
#5
f : C 6 N where C represents the colors and N represents the number of pellets dispensed
5o Celsius
A, B, and C are relations, but only A and B are functions.
Only G and J are functions.
Assignment 5.2
Problems
#1
The first three digits of a person's social security number is called the "area number." Before
1972, social security cards were issued in state offices, and the "area number" was assigned
according to the state where the card was issued.* This assignation of the "area number"
represents a function that maps a state to a three-digit number. Use descriptive notation to
describe the domain S and co-domain N of the function.
#2
The period in seconds of a pendulum operating at sea level is given by the formula below where L
equals the length of the pendulum in meters.
T = 2π
L
9.81
Use f : A → R notation to describe the function implied by the formula above. Be sure to
describe the domain and range of the function.
#3
Consider the formula that converts Celsius temperature readings to Fahrenheit and fill in the blank
below.
9
F = C + 32
5
The function f : C ° → F ° maps a Celsius reading of 3° to ______________________.
#4
Let X = Y = \ . Determine if the following subsets of X × Y are functions or relations.
F = {(1,1) , (1, 2 ) , (1, 3) , (1, 4 ) , (1, 5 ) , (1, 6 ) , (1, 7 )}
G = {(1,1) ,
( 2,1) , ( 3,1) , ( 4,1) , ( 5,1) , ( 6,1) , ( 7,1)}
H = {(1,1) , ( 2, 2 ) , ( 3, 3) , ( 4, 4 ) , ( 5, 5 ) , ( 6, 6 ) , ( 7, 7 )}
#5
Let X = Y = \ . Determine if the following graphs of H, G, J, and K (all subsets of X × Y ) are
functions or relations.
y
y
y
y
G
H
x
K
x
x
x
J
*
The "geographical code" contained in a social security number is not meant to be any kind of useable geographical
information. The numbering scheme was designed in 1936 (before computers) to make it easier for Social Security
Administration (SSA) to store the applications in their files in Baltimore since the files were organized by regions as well as
alphabetically. The area number is really just a bookkeeping device for the SSA's own internal use.