© Sanjay K. Chugh
5
Spring 2008
Chapter 1
Mathematical Refresher
The concept of a function is a very general and powerful one.
A function is a
mathematical object that serves as a fundamental tool in many fields of analysis.
We will
not here give a rigorous or comprehensive treatment of the mathematical notion of a
function.
The purpose here is to (re)familiarize you with the basic concepts and the most
important ways in which we will use functions as we develop tools of economic analysis
in this course.
Abstract Functions and Functional Forms
A
function
transforms an
input
into an
output.
More specifically, a function is a rule
that specifies how an input is to be transformed into some output.
At its simplest level,
the level with which we will be concerned, the input and outputs will all be numbers.
In
general, any function can have multiple inputs and multiple outputs.
Every function that
we will use will have only one output – that is, a function whose operation results in only
one numeric value as its output.
However, we will regularly encounter functions that
have multiple inputs, in addition to functions that have simply a single input.
A function can be written and used in abstract form, as when we simply write and use the
function
()
f
x
without specifying anything further about what the function actually does.
Often, however, in order to do something useful with a function, we need to specify a
particular
functional form –
that is, we often need to specify what a function actually
does (i.e., what the rule is).
Some examples of common functional forms will help
illustrate the concept:
2
f
xx
(
1
.
1
)
() 2 8
f
±
(
1
.
2
)
f
(
1
.
3
)
() l
n
f
(
1
.
4
)
(, ) l
n
() 0
.
8
l
n
f xy
x
y
±
(
1
.
5
)
In the above simple functions (functional forms), note that each function returns only one
number as its output (as promised).
Also note that function (1.5) is a function of two
inputs, while the others are all functions of one input.
Arguments of Functions
To be a bit more formal mathematically, an input(s) to a function is commonly known as
its
argument(s),
and the output of a function is commonly known as its
result
or
value.
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6
Spring 2008
Using the functions defined above, we see that each of the functions (1.1) through (1.4)
takes one argument named
x
.
Function (1.5) takes two arguments named
x
and
y
.
When actually performing numerical calculations using functions, the
x
in each case of
the functions (1.1) through (1.4) would be replaced with an actual number because it is
meaningless to
square the letter x,
because only
numbers
can be squared.
The leads to
the distinction between
formal arguments
and
actual arguments.
Think of a formal argument as a placeholder in an abstract function.
In each function
(1.1) through (1.4), the formal argument is
x
.
In function (1.5), the two formal
arguments are
x
and
y
.
More will be said about replacing formal arguments with actual
arguments, but first let’s examine the components of a function definition.
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 Spring '08
 chugh
 Optimization, Mathematical optimization, Constraint, lagrange multipliers, Sanjay K. Chugh

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