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Chapter 1: Functions and
Limits
1.1 Functions
We will think of functions in this course as inputoutput machines: a number
goes in, the machines does something to it, and it spits out another number.
H
H
±
±
H
H
±
±
input
output
Since the input and output can take on diﬀerent values they are
variables
. The value
of the output is determined by the value of the input. In other words, if the value of
the input is known there is only one possible value for the output. For this reason
the input is called the
independent variable
and the output is called the
dependent
variable
.
Notations for Functions
Although there are many ways to denote functions, we will restrict ourselves,
primarily, to two diﬀerent ways. Most of the other ways are mixtures of these two
ways.
Function notation:
When using function notation, we label the
process
(in other
words the box). When there is no other compelling name, we will often label it
f
or
g
.
H
H
±
±
H
H
±
±
f
Using this label, the output can then be expressed in terms of the inputs. For
instance, if the input is 2, the output is labeled
f
(2). If the input is
x
the output is
f
(
x
). If the input is

x
2
+ 3 then the output is
f
(

x
2
+ 3).
HH
±±
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±±
2
f
(2)
f
HH
±±
HH
±±
x
f
(
x
)
f
1
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View Full Document HH
±±
HH
±±

x
2
+ 3
f
(

x
2
+ 3)
f
This is generally the notation that we use when we are doing theory and don’t have
a particular application in mind.
Example:
Consider the function
f
given by
f
(
x
) =
πx
2
. Notice that this formula
describes the function. In other words it describes how the output is obtained from
the input; the input is squared and then multiplied by
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This note was uploaded on 05/06/2008 for the course MATH 118x taught by Professor Vorel during the Spring '07 term at USC.
 Spring '07
 Vorel
 Calculus, Limits

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