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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{Functions}

Calculus deals almost always with {}``functions{}''. Although the term has
different meanings in other situations, in mathematics a \emph{function} \( y=f(x) \),
or just \( f \), is always defined as the assignment, to each \( x \) in a
set called the \emph{domain} of the function, a value \( y \) in what is called
the \emph{range} of the function. To each \( x \) there is one and only one
\( y \). To indicate the domain \( A \) and the range \( B \) of \( f \)
(we often leave out the \( (x) \) when we refer to a function), we sometimes
write \( f:A\rightarrow B \). 

The reason we want to leave out the \( (x) \) of a function \( f \), calling
it by just the letter \( f \) rather than \( f(x) \), is sometimes confusing.
Let's say we define a function \( f \) by a formula, say \( f(x)=x^{2} \).
That means, to each number \( x \), this function assigns its square. You write
a particular case of that, say the value when \( x=2 \) (that value is 4, of
course), by writing \( f(2)=4 \). Really, we should write \( f(\, ) \) for
the function itself, but we leave out the parentheses. 

We can also replace the \( x \) by another letter, so for this same function
\( f(x)=x^{2} \), \( f(t) \) makes sense, and \( f(t)=t^{2} \). 

There is some confusion between the function \( f \) itself and its \emph{graph}.
The \emph{graph} of a function \( f(x)=x^{2} \), for \( x \) in a specified
domain (say from -1 to 2), is the set of points \( (x,f(x)) \) in the plane
where \( x\in [-1,2] \), that is, the set of points \( (x,y) \) where \( x\in [-1,2] \)
and for each such \( x \), the \( y \) is that one \( y \) satisfying \( y=f(x)=x^{2} \).
Here is the picture:

\vspace{0.3cm}
{\par\centering \resizebox*{2in}{2in}{\rotatebox{270}{\includegraphics{21-11-x^2.ps}}} \par}
\vspace{0.3cm}

One thing to note is that the graph satisfies the {}``vertical line test{}'',
that is, any vertical line meets the graph only once. 

Depending on the information available, a function can be described in a number
of different ways. The current text lists four distinct ways, but with a little
imagination you could probably come up with more. You've already seen functions
defined by formulas (or \emph{algebraically}), and graphs. A function can also
be described verbally, such as the following: For each \( t \), \( f(t) \)
is the \( x \)-coordinate of the point on the circle of radius 1, centered
at the origin, at an angle \( t \) from the positive \( x \)-axis. In this
case, \( f(t)=\sin (t) \). This verbal description may seem difficult (it was,
in a sense, meant to be), but if you look at the picture you see that, indeed,
\( \sin (t) \) is exactly the \( x \)-coordinate of the point on the unit
circle at an angle \( t \) from the positive \( x \)-axis.

\vspace{0.3cm}
{\par\centering \includegraphics{sine-definition.ps} \par}
\vspace{0.3cm}

A common way to describe a function in science and engineering is as a table
of values, that is, for various values of \( x \), values of \( f(x) \) are
explicitly written out.

Another way that functions are often defined for calculus texts, which you rarely
see elsewhere, is {}``piecewise{}'', that is, for some part of the domain
\( f \) is defined by one formula, and in other parts of the domain \( f \)
has another formula. We write these functions with a brace, indicating which
formula is used for which parts of the domain. An example would be:\[
f(x):=\left\{ \begin{array}{ll}
1-x, & \textrm{if}\, x\leq 1\\
x^{2}, & \textrm{if}\, x>1.
\end{array}\right. \]
Some of these {}``piecewise{}'' functions you have seen before, such as the
absolute value function \( g(x)=|x| \). Since \( |x| \) is the same as \( x \)
for \( x\geq 0 \), but \( |x| \) is the opposite sign for \( x<0, \) (erase
the negative sign by adding another) so \[
g(x)=|x|:=\left\{ \begin{array}{ll}
x, & \textrm{if}\, x\geq 0\\
-x, & \textrm{if}\, x<0.
\end{array}\right. \]
piecewise-defined functions allow us to think about a lot of properties of functions
that you don't see when you only deal with functions defined by a single formula.


\subsection*{More properties of functions}

Some functions have special properties that make them easier to understand.
The first notion is that of \emph{symmetry} (even or odd functions). A function
\( f(x) \) is \emph{even} if \( f(-x)=f(x) \), and is \emph{odd} if \( f(-x)=-f(x) \).
An even function has mirror symmetry with respect to reflection about the \( y \)-axis,
and an odd function has {}``flipped{}'' symmetry, or is symmetric with respect
to the origin.

\vspace{0.3cm}
{\par\centering \includegraphics{even-odd.ps} \par}
\vspace{0.3cm}

There is some reason to this terminology, since \( f(x)=x^{4} \) is an even
function, and \( g(x)=x^{3} \) is an odd function.
\end{document}
