One of the most important concepts in algebra is that of the function. The formal mathematical definition of the term function14 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:
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.
is the function name represented symbolically and
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
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:
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:
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:
Many functions do not have a unique inverse, however. For example, the function:
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:
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:
This leads us naturally enough into our next section (so keep it in mind) but first we have to deal with several important ideas.