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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Modeling}

You need to be familiar with a number of general classes of functions, functions
which occur as models of various natural phenomena


\subsubsection*{Linear~functions }

These are functions of the sort \( f(x)=ax+b \). Their graphs are lines.


\subsubsection*{Polynomials }

These are important primarily because we can understand their properties quite
easily, and because any function can be pretty-well approximated by a polynomial
for a limited range of values of \( x \) (more about that later). A \emph{polynomial
function of degree} \( n \) is a function of the sort \( f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0} \).
For example, \( f(x)=3x^{2}-2x+4 \) is a polynomial of degree 2.


\subsubsection*{Powers }

Functions of the sort \( f(x)=x^{a} \), where \( a \) can be any number. The
big thing to worry about here is to keep straight the rules for exponents, such
as 
\[
x^{1/2}=\sqrt{x},\]
 and 
\[
x^{-3/2}=\frac{1}{\left( \sqrt{x}\right) ^{3}}.\]



\subsubsection*{Trigonometric~functions }

Trigonometric functions come up in many different situations. Any model with
a geometric ``piece'' will often have something to do with trigonometric functions.
In particuar, anything to do with a circle, or triangles, or repeted ``cycles''
of motion will use trigonometric functions somewhere. You should know the basic
properties of functions like \( \sin (x) \), \( \cos (x) \), \( \tan (x) \),
etc., where \( x \) is an angle in radians.


\subsubsection*{Exponential~functions }

These are functions of the sort \( f(x)=a^{x} \). Note that the \( x \) is
in the exponent. The number \( a \) is the \emph{base}. We'll introduce these
later, but again we expect you to be familiar with their basic properties. 

The primary reason to study exponential functions is that they model many natural
phenomena, that is, the behavior of many systems are expressed in terms of exponential
functions. Such things as compound interest, radioactive decay, and population
growth are governed by exponential functions.

One curious number always mentioned in this context is \( e \). \( e \) is
just a number, like \( \pi  \), which has a specific value. That letter, \( e \),
is almost-always reserved for that particular number (again, like \( \pi \approx 3.14159 \).
\( e \) is approximately \( 2.71828 \). But \( e, \) like \( \pi  \), comes
from a specific relationship. If you look at the graphs of the various exponential
functions \( y=a^{x} \) for a number of choices of \( a \), they all pass
through \( (1,0) \), since \( a^{0}=1 \) for any positive number \( a \).
\( e \) is that choice of base so that the slope of the curve as it passes
through \( (1,0) \) is 1. 


\subsubsection*{Logarithms }

These are the inverses of exponential functions, written \( f(x)=\log _{a}x \).
Again, \( a \) is the \emph{base}. 

The text mentions ``transcendental functions'' as if they were a specific
class of functions. But they are not. A function is algebraic if it is a sum
or quotient of polynomials and roots (fractional powers). If a function is not
algebraic, it is \emph{transcendental}.


\subsection{Combining functions together}

You should be familiar with how to add functions together, \( f+g(x)=f(x)+g(x) \),
as well as multiplication and division of functions. The biggest concern will
be composition of functions, one function followed by another. That is \( f(g(x)) \)
takes \( x \), does \( g \) to it, and then does \( f \) to the result. You
will use this most often to recognize that a function, such as 
\[
\sin (\sqrt{3x+2}),\]
 is a composition of simpler functions, and so can be analyzed by breaking it
down into those simpler components.

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