Website owner: James Miller
GRAPHICAL INTERPRETATION OF REAL-VALUED FUNCTIONS
There is a simple graphical interpretation of a real-valued function y = f(x) of a single real variable x. It is a curve as shown in Fig. 1 obtained by plotting the values of y at different values of x. In the case of a real-valued function z = f(x, y) of two real variables x and y we find a graphical interpretation by going to three dimensions. We view the function as a surface obtained by erecting elevations corresponding to values of z at different points (x, y). See Fig. 2. However, when we go to a real-valued function u = f(x, y, z) of three real variables x, y and z we are unable to think of any similar graphical interpretation. Let us consider another very useful way of viewing a function, a way that can be used, not only in the case of two variables, but also in the case of three or more variables. We will illustrate it using the case z = f(x, y) of two variables.
Let us say that the function z = f(x, y) is defined over some region R in the xy-plane. Now imagine in your mind that to each point in region R there is assigned a number equal to the value of the function at that point. See Fig. 3. Now drawn in lines connecting points of equal value for various selected values as shown in Fig. 4. The lines of equal value are called level lines or level curves. In the figure the different level curves correspond to the equations
f(x, y) = 5
f(x, y) = 10
f(x, y) = 15
What we have here is completely analogous to the contour maps of topography where the contour lines represent lines of equal elevation. The isobars showing curves of equal atmospheric pressure on a meteorological chart are another example.
So the technique is as follows: Choose a sequence of numbers z1, z2, ... , zn representing parameterized values of the function f(x, y) and then draw the curves
f(x, y) = z1
f(x, y) = z2
f(x, y) = zn .
Each curve then represents a level curve of the function z = f(x, y). Using this technique one can form a good idea of the variation the function f(x, y) just as one can get a good idea of the lay of terrain from the contour lines of a topographic map.
In this same way we can visualize a function of three variables, u = f(x, y, z), by choosing a sequence of numbers u1, u2, ... un and considering the set of level surfaces
f(x, y, z) = u1
f(x, y, z) = u2
f(x, y, z) = un .
in three dimensional space.
In the case of a function with n variables
u = f(x1, x2, ..., xn)
we can envision the function as consisting of a set of level surfaces in n-space. We parameterize u, giving it values u1, u2, ... un, and consider the surfaces
f(x1, x2, ..., xn) = u1
f(x1, x2, ..., xn) = u2
f(x1, x2, ..., xn) = um .