Many-valued functions

Many-valued functions

Single valued function

It is characteristic of all the graphs of elementary single value functions, that perpendiculars to the axis X cut the graph in not more than one, and for the most part in one point. This means that, given x in the function defined by the graph, one corresponding value of y is defined. A function of this sort is said to be single-valued.

If perpendiculars to X cut the graph in more than one point this means that, given x, there are several corresponding ordinates of the graph, i.e. several values of y. Such functions are called many-valued; they have already been mentioned in the concept of function.

Although the direct function y=f(x) is single-valued, its inverse can be many-valued. This is evident, for instance, from Fig. 25.

We shall consider in detail one elementary case. The continuous curve of Fig. 13 is the graph of the function y = x2. If the figure. 16 is turned through 180° about the bisector of the first quadrant of the axes, the graph of the inverse function y = ; is obtained (Fig. 26).

Let us consider it in detail. For negative x (to the left of axis Y), perpendiculars to axis X do not intersect the graph, i.e. the function y = is not defined for x < 0. This corresponds to the fact that the square root of a negative number has no real value. For positive x, however, perpendiculars to X cut the graph in two points, i.e. for a given positive x we have two ordinates of the graph: MN and MN1 The first ordinate gives a certain positive y, the second a negative value of the same absolute magnitude. This corresponds to the fact that the square root of a positive number has two values, equal in absolute magnitude and opposite in sign. It is also evident from the figure that we have only one value y = 0 for x = 0. Thus, the function y = , defined for x > 0, has two values for x > 0, and one value for x = 0.

It may also be noted that the part of the graph of the function y = , shown in Fig. 27, is obtained from the part of the graph of the direct function y = x^2 (Fig. 13) lying on the right of axis Y. The part of the graph of

lying in the first quadrant of the axes, has already been illustrated in Fig. 22.

We now turn to the case when rotation of a single-valued direct function leads to a single-valued inverse function. A new concept must be introduced for this.

The function y = f(x) is said to be increasing, if y increases when the independent variable x increases, i.e. if x2 > x1 implies f(x2) >f(x1).

With the axes X, Y as used by us, increasing x implies movement along X to the right, and increasing y, movement upwards along Y. It is characteristic of the graph of an increasing function that, on moving along the curve in the direction of increasing x (to the right), we also move in the direction of increasing y (upwards). Let us consider the graph of a single-valued increasing function, defined in the interval a < x < b (Fig. 28). 

Let f(a) = c and f(b) = d, so that evidently, since the function is increasing, c < d. If we take any y in the interval c < y < d, and draw the corresponding perpendicular to the axis Y, this perpendicular will cut our graph in only one point, i.e. for every y in the interval c < y < d there is one corresponding definite value of x. In other words, the inverse of an increasing function is single-valued.

It is clear from the figure that this inverse function is increasing. Similarly, the function y = f(x) is said to be decreasing, if increase of the independent variable x implies decrease of the corresponding y, i.e.

if x2 > x1 implies f(x1) > f(x2) · It can be shown, as above, that the inverse of a decreasing function is a single-valued, decreasing function.

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek