Examples of Functions

In a compact form a function is a well defined unidirectional dependency, a concept of which I shall provide several examples.

Functions that are not numeric

  • The set of fingerprints is uniquely defined for every person. That is to say, there is a function (call it $f)$ from the set of people to the set of fingerprint sets. The function is not defined for every person, but only for those who were fingerprinted. Any fingerprint set uniquely defines a person. This function is inverse to $f.$

  • For every person, there is a unique DNA molecule whose copies are carried by every cell in human body. Thus there is a function from the set of people to the set of molecular structures known as the DNA. This function does not have an inverse; for, say, a pair of identical twins share the DNA structure.

  • Every triangle has the barycenter, the incenter, the orthocenter, the circumcenter, the symmedian point, the isogonic center and many other remarkable points. These all are various functions defined on a set of triangles. Every triangle center is defined by a homogeneous function of in, say, barycentric coordinates, which is symmetric with respect to its arguments.

  • Every point in a plane of a Mandelbrot set defines a Julia set.

  • I would like to claim that, for every object, its optimal appearance occurs at a certain distance, which would make this distance a function of the appearance. But this may not be true: an object may look equally well from several distinct distances.

  • In a topological space, every set has a boundary, an interior, and a closure. The correspondence defines three functions on the set of all subsets of a topological space.

  • Any coloring in a game of Y can be uniquely reduced to a coloring of a smaller board in a manner that preserves the winning chains.

Function addition and various function multiplications serve additional examples of functions of two variables.

Numeric functions

The distinction between numeric and non numeric function is rather nebulous. It's neither standard nor common. I would think of a function as numeric provided that, on top of establishing a correspondence between number sets, the function gains some properties from that fact. If the numbers are used merely as tags or convenience symbols, house numbers for example, there is probably no point in thinking of the function as numeric. The house numbers may still be used to indicate the proximity of a house from a point of departure. More generally, we talk of a distance function, which I feel comfortable to think of as numeric.

The most pedestrian definition of a function is the one common in high school textbooks, see for example [Jacobs, p. 122]:

A function is a pairing of two sets of numbers so that to each element in the first set there corresponds exactly one number in the second set.

So, what makes a function numeric? As I already mentioned, the distinction is loose, but mostly one thinks of a function as numeric if it is defined by means of an algebraic formula. A more broad convention that only requires the two paired sets in the definition to consist of (whatever) numbers, allows functions that only nominally relate numbers to one another and do not gain any essential properties from their being defined for number sets.

  • The function $x\mapsto x^2$ relates a number to its square. We would commonly write $f(x) = x^2,$ but, to denote the result as, say, $^{2}(x)$ is as consistent with Euler's notations as could be. In fact, for the binomial coefficient "n choose k", which is a function of two variables, several different notations - $C(n, k),$ $_{n}C_{k},$ $\displaystyle C^{n}_{k}$ - are in common use.

  • A more general quadratic function is defined by

    $f(x) = ax^{2} + bx + c$

    and is the second simplest of the polynomial functions. For the simplicity title, I think, compete two functions: the constant function $f(x) = const,$ and the identity function $f(x) = x.$

  • Some function, because of their importance and frequent use in mathematics and applications, have, with time, been granted special names. Such are the trigonometric functions $\sin (x),$ $\cos (x),$ and others, exponential function $a^{x}$ and the logarithm $\log (x).$ Having a name does not mean that the values of the function can be easily calculated. Virtually for all x, sin(x) has to be approximated. But the same is true for apparently simpler functions, like $x^{1/2}=\sqrt{x}.$

  • New functions can be constructed as the combinations (in many senses) of other functions. Some combinations yield surprising results. For example, incomprehensible at first sight expression

    $x + \arctan (\cot\pi \cdot x))/\pi - 1/2$

    is just a representation of the floor function:

    $\lfloor x\rfloor = x + \arctan (\cot\pi \cdot x))/\pi - 1/2.$

    (I thank Andrew Newton for this example and for the link to an interesting discussion.)

    In a similar vein, Mohamed Al-Dabbagh who discovered the representation for the floor function in 1996, has also found that

    $|x| = \arccos (\cos\pi \cdot x/(1 + x^{2})))\cdot (1 + x^{2})/\pi$ ,
    $\mbox{sign}(x) = \arccos (\cos\pi \cdot x/(1 + x^{2})))\cdot (1 + x^{2})/\pi \cdot x,$
    $H(x) = (\mbox{sign} (x) - 1)^{2}/4,$

    where $H(x)$ is the Heaviside Step Function: $H(x) = 1,\,1/2,\,0$ depending on whether $x \gt 0,$ $x = 0,$ or $x \lt 0.$

  • Natural as it appears to be and common as it is, using formulas to define functions may be quite treacherous. Iain Stewart tells about his experience with offering the students to find the derivative of the function

    $f(x) = \log (\log (\sin(x))).$

    By applying standard rules of the calculus, most students derive the following answer:

    $f' = \cot (x)/\log (\sin(x)).$

    Curiously, the derivative makes sense for the values of $x$ where $\sin (x) \gt 0.$ But having said that, we should also note that $\log (\sin(x))$ is never positive, so that $\log (\log (\sin(x)))$ does not make sense for any $x.$

References

  1. H. R. Jacobs, Mathematics: A Human Endeavor, 3rd edition, Freeman, 2002 (6th printing)
  2. I. Stewart, Concepts of Modern Mathematics, Dover, 1995 (3rd printing), pp. 64-65

Functions

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