 Chris Fox's Engineering Section

Mathematical Functions

In mathematics, a function shows the relationship between two or more variables. For example, the expression "y is a function of x" may be summarised as either y=f(x) or y=y(x). In this example, y is known as the dependent variable and x is known as the independent variable.

The domain of the function includes all the values of the independent variable on which the function may be performed, and the range includes all the values of the dependent variable that may be produced as a result. In the case of y=mx+c (The equation for a straight line), the domain and range both run from -∞ to ∞.

The relationship between two variables can be shown either by writing an equation or by drawing a graph. I will describe one method of drawing a graph, using Cartesian co-ordinates. Another system, using polar co-ordinates, is outlined on the Trigonometry page.

Two lines - termed axes - are drawn at right angles to each other, the x-axis running from left to right and the y-axis running upwards. The point where they meet is called the origin and has the co-ordinates (0,0). The co-ordinates of a point on the graph are given relative to the origin, the x co-ordinate being given first; and the positive x and y directions are right and up respectively. So by moving a units to the right from the origin then b units up, one arrives at the point (a,b).

One can now use either an equation or experimental data to plot points (loci) on the graph. To take the example of the straight line, if b=ma+c the point (a,b) is a locus of the function, so a point can be plotted there. Then do the same for other loci and draw a line through the points.

A graph can be used to show the relationship between any pair of variables, provided such a relationship exists. The simplest name of the graph gives the variable on the y-axis followed by the variable on the x-axis, e.g. a speed-time graph.

An even function is one for which f(-x)=f(x). It will be seen that on a graph, such a function has reflection symmetry about the y-axis. For an odd function, f(-x)=-f(x). On a graph, such a function has two-fold rotation symmetry about the origin.

An equation of the form y=f(x) can be rearranged to make x the subject of the formula. The result is an equation of the form x=f-1(y), where f-1() denotes an inverse function. As an example, let us rearrange y=1/(1-x). Multiplying both sides by (1-x)/y gives 1-x=1/y, and adding x-1/y to both sides of this gives 1-1/y=x. The inverse of 1/(1-x) is therefore 1-1/x.

If Cartesian co-ordinates are extended to three dimensions, the third axis is termed the z-axis. This axis runs perpendicular to the other axes and meets them at the origin, and the positive z direction is towards the viewer.