Chapter 2
Review of Simple Functions
In this chapter we review a few basic concepts related to functions, and introduce the simplest family
of functions that have interesting, nontrivial properties: the power functions. We explore geometric
and graphical properties of these, commenting on special attributes that will become important
in later study.
Using the power functions as basic building blocks, we construct the family of
polynomials, and investigate how their features are inherited from the underlying behaviour of
power functions.
Here, we begin to develop a few important curvesketching skills.
We end the
chapter with applications of these concepts to examples from biology.
2.1
What is a function
A function is just a way of expressing a special relationship between a value we consider as the
input (“
x
”) value and an associated output (
y
) value. We write this relationship in the form
y
=
f
(
x
)
to indicate that
y
depends on
x
. The only constraint on this relationship is that, for every value of
x
we can get at most one value of
y
. This is equivalent to the
“vertical line property”
: the graph
of a function can intersect a vertical line at most at one point. The set of all allowable
x
values is
called the
domain
of the function, and the set of all resulting values of
y
are the
range
.
Naturally, we will not always use the symbols
x
and
y
to represent independent and dependent
variables. For example, the relationship
V
=
4
3
πr
3
expresses a functional connection between the radius,
r
, and the volume,
V
, of a sphere. We say in
such a case that “
V
is a function of
r
”.
All the sketches shown in Figure 2.1 are valid functions.
The first is merely a collection of
points,
x
values and associated
y
values, the second a histogram. The third sketch is here meant to
represent the collection of smooth continuous functions, and these are the variety of interest to us
here in the study of calculus. On the other hand, the example shown in Figure 2.2 is not the graph
of a function. We see that a vertical line intersects this curve at more than one point. This is not
permitted, since as we already said, a given value of
x
should have only one corresponding values
of
y
.
v.2005.1  September 23, 2009
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Math 102 Notes
Chapter 2
x
x
x
y
y
y
Figure 2.1: All the examples above represent functions.
x
y
Figure 2.2: The above elliptical curve cannot be the graph of a function. The vertical line (shown
dashed) intersects the graph at more than one point: This means that a given value of
x
corresponds
to ”too many” values of
y
. If we restrict ourselves to the top part of the ellipse only (or the bottom
part only), then we can create a function which has the corresponding graph.
2.2
Geometric transformations
It is important to be able to easily recognize what happens to the graph of a function when we
change the relationship between the variables slightly. Often this is called
applying a transformation
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '09
 Allard
 Review of Simple Functions

Click to edit the document details