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Economics 214
Lecture 3
Introduction to Functions
Variables
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Variables studied in economics can be
qualitative or quantitative.
Qualitative variable represents some
distinguishing characteristic, such
male or female, employed or
unemployed.
Quantitative variables can be
measured numerically.
Numbers
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Integers are whole numbers.
Real numbers include all integers and all
numbers between the integers.
Numbers that can be expressed as ratios of
integers are call rational numbers.
Numbers that cannot be expressed as ratios
of integers are call irrational numbers.
Intervals
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Interval is set of all real numbers between to
endpoints.
Closed interval includes the endpoints. i.e. [1,2]
Open interval between two numbers excludes the
endpoints. i.e. (1,2)
Half-closed or half-open interval between 2
numbers includes one endpoint and excludes the
other endpoint. i.e. (1,2]
Infinite interval has negative infinity, positive
infinity or both as endpoints. i.e. [0,∞)
Sets
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Set is simply a collection of items.
Item included in a set are called
elements.
C={freshman,sophmore,junior,senior}
To show item is part of set we use
symbol, . i.e. freshmanC
To show item is not part of set, we use
symbol, . i.e. graduate studentC.
Sets
A set can be described either by listing all its
elements or by describing the conditions
required for membership. For Example
N={10,20,30,40}
Or
N={x|x=10*y, y=1,2,3,4}
Relations
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The elements of one set can be associated
with the elements of another set through a
relationship.
A function is a relationship that has a rule
that associates each element of one set with
a single element of another set.
A function is also called a mapping or a
transformation.
Function
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A function f that unambiguously
associates with each element of a set
X one element in the set Y is written as
f:XY.
The set X is called the domain of the
function f.
The set of values that occur is called
the range of the function f.
Example Function
X={1,2,3,4}
f:Y=10X
Y={10,20,30,40}
f:XY
Univariate Function
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A Univariate function maps one number, a
member of the domain, to one and only one
number, element of the range.
We represent the univariate function as
y=f(x).
Y is the dependent variable or value of the
function.
x is the independent variable or argument
of the function.
Examples of Univariate
functions
f ( x) can represent any relationsh ip that
maps one x to one y.
2
y
x0
x
2
y    x  x
Ordered Pairs
An Ordered pair is two numbers presented in
parentheses and separated by a comma, where
the first number represents the argument of the
function and the second number represents the
corresponding value of the function. Each
ordered pair for the function y=f(x) takes the form
(x,y).
Example of Ordered Pair
2
We will evaluate our function y 
for
x
x  1,2,4,8,16
(1,2) (2, 2 ) (4,1) (8,1
2 ) (16,0.5)
Graphing
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Ordered pairs can be plotted in a
Cartesian plane.
The origin of the plane occurs at the
intersection of the two axes that are a
right angles to each other.
Points along the horizontal axis
represent values of the argument of
the function.
Graphing Continued
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Points along the vertical axis represent
values of the function.
The coordinates of a point are the values of
its ordered pair and represent the address
of that point in the plane.
The x-coordinate of the pair (x,y) is called
the abscissa, and the y-coordinate is called
the ordinate.
The origin is represented by the ordered
pair (0,0).
Plot of our function
y=2/(x)^0.5
2.5
2
y
1.5
y
1
0.5
0
0
5
10
x
15
20
Graph
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Graph of a function represents all
points whose coordinates are ordered
pairs of the function.
Graph of our function
y=2/sqrt(x)
2.5
2
y
1.5
y
1
0.5
0
0
5
10
x
15
20
Linear function
A linear function t akes the form
y    x
 is the intercept of the function, the value of
the function w hen the argument equals 0.
 is the slope of the function. Represents the change
in the value of the function associated with a given
change in its arguments.
 f ( xB )  f ( x A )    xB     x A 



xB  x A
xB  x A


Graph Linear Function
y=2+0.5x
14
y
12
10
8
6
y
4
2
0
0
5
10
15
x
20
25
Graph function in multiple
quadrants
y=-5-2x+0.3x^2
20
15
10
5
y
y
0
-10
-5
-5 0
5
-10
x
10
15