“I am a function”

The experimenter can move back and forth along a 4-meter track. His position is detected by a photocell. The graph of a given function appears on a screen in a coordinate system. The experimenter is now required to “trace” this curve by his movements in the form of a path-time diagram. The experiment takes 10 seconds.

The experiment will now be explained by means of two examples:

If the given path-time-diagram (e.g. for 6 seconds) has the following form

Figure 1: First curve run

this means: You start directly in front of the screen and then move in the following way:

  • In phase 1, one moves away from the screen at a constant speed.
  • In phase 2, you move away from the screen with increasing speed.
  • In phase 3 you stop; the distance to the screen remains constant.
  • In phase 4, you move away from the screen (again) at a constant speed.
  • In phase 5, one moves away from the screen with decreasing speed.
  • In phase 6 you stop.

If the path-time diagram given as a running rule has the following form

Figure 2: Second curve run

then this means

  • Start (at time t=0) at a distance of three meters from the screen.
  • For 0\lt t\lt 1 (i.e. within the first second), approach the screen to a distance of 2.91m.
  • For 1\lt t\lt 7 (i.e. within the next 6 seconds) move first (t\lt 4) with increasing then (t\gt 4) with decreasing speed towards a distance of 3.82m from the screen.
  • For t\gt 7 you run again with increasing speed up to 2.91m towards the screen.

Here, the time is specified in seconds.


And now … the mathematics of it:

In mathematics, a function f (or mapping) is a relation between two sets A and B that assigns to each element x (also independent variable or x-value) of A (domain of definition) exactly one element y (also dependent variable or y-value) of B (domain of values).

The concept of a function or figure occupies a central position in modern mathematics. In a so-called path-time diagram, the function under consideration f\colon t\mapsto y=f(t) is a function of time t (t\gt 0) with real values.

In the experiment “I am a function” these values of the function f are between 0 and 4, indicating the possible distance (in meters) of the visitor from the screen. The definition range is A={t\,|\,t\gt 0} and the value range B={s\,|\,0\lt s\lt 4}.

But not only in mathematics, also in everyday life the notion of function plays a very important role. For example, numerous processes, such as movements, temperature and pressure curves, and economic processes can be described by functions.

There are different formulations for describing functions, as the following example shows:

  • Function term:

        \[f(x)=x^2+1.\]

  • Function equation:

        \[y=x^2+1.\]

  • Assignment rule:

        \[x\mapsto x^2+1.\]

Each real number x is assigned its square x^2 increased by the value 1, i.e. x^2+1. The graphical representation in a rectangular (Cartesian) coordinate system then has the following form:

Figure 3: Graph of the function y=x^2+1

For finite, but also countably infinite definition ranges, it is also possible to specify a function by a table of values, such as for y=f(x)=x^2+1 for discrete (countably infinite) values x=1,2,3,\ldots

x1234567\ldots
y=x^2+1251017263750\ldots
Table 1: Table of values of the function y=x^2+1

Literature

[1] Beutelspacher, A.: Mathematik zum Anfassen, Gießen, 2005.

[2] Lehmann, I., und Schulz, W.: Mengen — Relationen — Funktionen: Eine anschauliche Einführung, 3. Auflage, Wiesbaden, 2007.

[3] Nollau, V.: Mathematik für Wirtschaftswissenschaftler, 4. Auflage, Wiesbaden, 2003.

[4] Warlich, L.: Grundlagen der Mathematik für Studium und Lehramt: Mengen, Funktionen …, Wiesbaden, 1996.