One of the most important concepts in algebra is that of the *function*. The formal mathematical definition of the term
function^{14} is beyond the scope of this
short review, but the summary below should be more than enough to work
with.

A function is a *mapping* between a set of *coordinates* (which
is why we put this section *after* the section on coordinates) and a
*single value*. Note well that the ``coordinates'' in question do
*not have to be space and/or time*, they can be any set of *parameters* that are relevant to a problem. In physics, coordinates can
be any or all of:

- Spatial coordinates,
- Time
- Momentum
- Mass
- Charge
- Angular momentum, spin, energy, isospin, flavor, color, and much
more, including ``spatial'' coordinates we
*cannot see*in exotica such as string theories or supersymmetric theories.

Note well that many of these things can *equally* well be functions
themselves - a potential energy function, for example, will usually
return the value of the potential *energy* as a function of some mix
of spatial coordinates, mass, charge, and time. Note that the
coordinates can be continuous (as most of the ones above are
classically) or *discrete* - charge, for example, comes only
multiples of
and color can only take on three values.

One formally denotes functions in the notation e.g.
where
is the *function name* represented symbolically and
is
the entire vector of coordinates of all sorts. In physics we often
learn or derive functional forms for important quantities, and may or
may not express them as functions in this form. For example, the
kinetic energy of a particle can be written either of the two following
ways:

(32) | |||

(33) |

These two forms are equivalent in physics, where it is usually ``obvious'' (at least when a student has studied adequately and accumulated some practical experience solving problems) when we write an expression just what the variable parameters are. Note well that we not infrequently use

(34) |

is a function of , and but includes the gravitational constant N-m /kg in symbolic form. Not all symbols in physics expressions are variable parameters, in other words.

One important property of the mapping required for something to be a
true ``function'' is that there must be only a *single value* of the
function for any given set of the coordinates. Two other important
definitions are:

**Domain**- The
*domain*of a function is the set of all of the coordinates of the function that give rise to unique non-infinite values for the function. That is, for function it is all of the 's for which is well defined. **Range**- The
*range*of a function is the set of all values of the function that arise when its coordinates vary across the entire domain.

Two last ideas that are of great use in solving physics problems
algebraically are the notion of *composition* of functions and the
*inverse* of a function.

Suppose you are given two functions: one for the potential energy of a mass on a spring:

(35) |

where is the distance of the mass from its equilibrium position and:

(36) |

which is the position as a function of time. We can form the composition of these two functions by substituting the second into the first to obtain:

(37) |

This sort of ``substitution operation'' (which we will rarely refer to by name) is an

With the composition operation in mind, we can define the inverse. Not
all functions have a unique inverse function, as we shall see, but most
of them have an inverse function that we can use *with some
restrictions* to solve problems.

Given a function
, *if* every value in the range of
corresponds to one and only one value in its domain
, then
is *also* a function, called the *inverse* of
. When
this condition is satisfied, the range of
is the domain of
and vice versa. In terms of composition:

(38) |

and

(39) |

for any in the domain of and in the range of are

Many functions do not *have* a unique inverse, however. For
example, the function:

(40) |

does not. If we look for values in the domain of this function such that , we find an

(41) |

for The mapping is then one value in the range to

We can get around this problem by *restricting the domain* to a
region where the inverse mapping *is* unique. In this particular
case, we can define a function
where the domain of
is only
and the *range* of
is restricted to
be
. If this is done, then
for all
and
for all
. The
inverse function for many of the functions of interest in physics have
these sorts of restrictions on the range and domain in order to make the
problem well-defined, and in many cases we have some degree of *choice* in the best definition for any given problem, for example, we
could use *any* domain of width
that begins or ends on an odd
half-integral multiple of
, say
or
if it suited the needs of our problem to do so
when computing the inverse of
(or similar but different ranges
for
or
) in physics.

In a related vein, if we examine:

(42) |

and try to construct an inverse function we discover two interesting things. First, there are two values in the domain that correspond to each value in the range because:

(43) |

for all . This causes us to define the inverse function:

(44) |

where the sign in this expression selects one of the two possibilities.

The second is that once we have defined the inverse functions for either
trig functions or the quadratic function in this way so that they have
restricted domains, it is natural to ask: Do these functions have any
meaning for the *unrestricted* domain? In other words, if we have
defined:

(45) |

for , does exist for

This leads us naturally enough into our next section (so keep it in mind) but first we have to deal with several important ideas.