In this section, we'll briefly cover a few of the most relevant and important classifications of functions.

Every function can either be classified as an even function, an odd
function, or neither. Even functions have the characteristic that
*f* (*x*) = *f* (- *x*)
. They are symmetrical with respect to the y-axis. A line segment joining
the points
*f* (*x*)
and
*f* (- *x*)
will be perfectly horizontal. Odd functions have
the characteristic that
*f* (*x*) = - *f* (- *x*)
. They are symmetrical with respect to
the origin. A line segment joining the points
*f* (*x*)
and
- *f* (- *x*)
always
contains the origin. Many functions are neither even nor odd.

Some of the most common even functions are
*y* = *k*
, where
*k*
is a constant,
*y* = *x*
^{2}
, and
*y* = cos(*x*)
. Some of the most common odd functions are
*y* = *x*
^{3}
and
*y* = sin(*x*)
. Some functions that are neither even nor odd include
*y* = *x* - 2
,
*y* =
, and
*y* = sin(*x*) + 1
.

Figure %: The function on the left is even; the function on the right is odd.
Note the different types of symmetry.

Among the types of functions that we'll study extensively are polynomial, logarithmic, exponential, and trigonometric functions. Before we study those, we'll take a look at some more general types of functions.

The inverse of a function is the relation in which the roles of the
independent anddependent variable are reversed. Let
*f* (*x*) = 2*x*
. The
inverse of
*f*
,
*f*
^{-1}
(not to be confused with a negative
exponent), equals
. It is written like this:
*f*
^{-1}(*x*) =
. The
inverse of a function can be found by switching the places of
*x*
and
*y*
in the
formula of the function. The inverse of any function is a relation.
Whether the inverse is a function depends on the original function
*f*
. If
*f*
is a one-to-one function, then its inverse is also a function. A one-to-one
function is a function for which each element of the range corresponds to
exactly one element of the domain. Therefore if a function is not a one-to-
one function, its inverse is not a function. The horizontal line test shows
us that if a horizontal line can be placed in a graph such that it intersects
the graph of a function more than once, that function is not one-to-one, and its
inverse is therefore not a function.

Inverse functions are important in solving equations. Sometimes the solution
*y*
to a function is known, but the input for that solution
*x*
is not known.
In situations like these, the inverse of the function can be used to find
*x*
.
We'll see more inverse functions later.

A piecewise function is a function which is defined by different rules
depending on the value of the independent variable. The following piecewise
function is graphed below:
*y* =
for
*x*≤ 0
,
*y* = *x*
for
0 < *x* < 2
, and
*y* = 2
for
*x*≥2
.

Figure %: A piecewise function is governed by different rules depending on the
value of the independent variable.

A periodic function is a function with the following characteristic:
*f* (*x*) = *f* (*x* + *c*)
, for all values of
*x*
, where
*c*
is some constant. This means that
the values of
*f*
repeat themselves in a regular cycle. The most common
periodic functions are the trigonometric functions, but others exist as well.