Functions

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:

• Spatial coordinates, $x,y,z$
• Time $t$
• Momentum $p_x,p_y,p_z$
• Mass $m$
• Charge $q$
• 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 $e$ and color can only take on three values.

One formally denotes functions in the notation e.g. $F(\Vec{x})$ where $F$ is the function name represented symbolically and $\Vec{x}$ 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:

$$K(m,\Vec{v}) = \frac{1}{2}m v^2$$ (32)

$$K = \frac{1}{2}m v^2$$ (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 non-variable parameters - in particular constants of nature - in our algebraic expressions in physics as well, so that:

$$U = - \frac{G m_1 m_2}{r}$$ (34)

is a function of $m_1, m_2$ , and $r$ but includes the gravitational constant $G = 6.67\times 10^{-11}$ N-m$^2$ /kg$^2$ 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 $f(x)$ it is all of the $x$ 's for which $f$ is well defined.

Range

The range of a function is the set of all values of the function $f$ that arise when its coordinates vary across the entire domain.

For example, for the function $f(x) = \sin(x)$ , the domain is the entire real line $x \in (-\infty,\infty)$ and the range is $f \in [-1,1]$ ¹⁵.

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:

$$U(x) = \frac{1}{2} k x^2$$ (35)

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

$$x(t) = x_0 \cos(\omega t)$$ (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:

$$U(t) = \frac{1}{2}k x_0^2 \cos^2(\omega t)$$ (37)

This sort of ``substitution operation'' (which we will rarely refer to by name) is an extremely important part of solving problems in physics, so keep it in mind at all times!

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 $f(x)$ , if every value in the range of $f$ corresponds to one and only one value in its domain $x$ , then $f^{-1} = x(f)$ is also a function, called the inverse of $f$ . When this condition is satisfied, the range of $f(x)$ is the domain of $x(f)$ and vice versa. In terms of composition:

$$x_0 = x(f(x_0))$$ (38)

and

$$f_0 = f(x(f_0))$$ (39)

for any $x_0$ in the domain of $f(x)$ and $f_0$ in the range of $f(x)$ are both true; the composition of $f$ and the inverse function for some value $f_0$ yields $f_0$ again and is hence an ``identity'' operation on the range of $f(x)$ .

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

$$f(x) = \cos(x)$$ (40)

does not. If we look for values $x_m$ in the domain of this function such that $f(x_m) = 1$ , we find an infinite number:

$$x_m = 2\pi m$$ (41)

for $m = 0, \pm 1, \pm 2, \pm 3...$ The mapping is then one value in the range to many in the domain and the inverse of $f(x)$ is not a function (although we can still write down an expression for all of the values that each point in the range maps into when inverted).

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 $g(x) = \sin^{-1}(x)$ where the domain of $g$ is only $x \in [-1,1]$ and the range of $g$ is restricted to be $g \in [-\pi/2,\pi/2)$ . If this is done, then $x = f(g(x))$ for all $x \in [-1,1]$ and $x = g(f(x))$ for all $x \in [-\pi/2,\pi/2)$ . 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 $\pi$ that begins or ends on an odd half-integral multiple of $\pi$ , say $x \in (\pi/2,3\pi/2]$ or $x \in [9\pi/2,11\pi/2)$ if it suited the needs of our problem to do so when computing the inverse of $\sin(x)$ (or similar but different ranges for $\cos(x)$ or $\tan(x)$ ) in physics.

In a related vein, if we examine:

$$f(x) = x^2$$ (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:

$$f(x) = f(-x)$$ (43)

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

$$g(x) = \pm x^{1/2} = \pm \sqrt{x}$$ (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:

$$g(x) = +\sqrt{x}$$ (45)

for $x \ge 0$ , does $g(x)$ exist for all $x$ ? And if so, what kind of number is $g$ ?

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