In many disciplines such as business, law and psychology, behavior of rational functions (functions that exhibit self-organization) can be studied mathematically as well as qualitatively. However, in the** behavior of rational functions** there are no significant numerical parameters so the study is qualitative. Quantitative analysis is based on comparing values of behavior of rational functions in terms of another quantitative variable. Thus for example, a comparison can be made between the behavior of a salesperson with that of an accountant or a lawyer.

In the mathematical case, we say that the frequency of x increases or decreases, while the mean value of the exponential function H(x) changes from some initial value to some final value over a period t, while the corresponding value of the gamma function H(t) changes with time t squared. Both the real and the imaginary polynomial functions can be analyzed using the binomial tree.

The binomial tree in **behavior of rational functions** consists of a polynomial equation whose solutions lie within the range of the original equation. Accordingly, by taking the root of the tree, all subsequent levels of the tree can be obtained by means of increasing solutions of the polynomial equations.

There are many different approaches to quantifying **behavior of rational function**s, some of which are mathematically precise but some of which are more approximate. One popular approach to positive infinity is represented by a straight line on the graph, representing the set of real numbers, while the other approach negative infinity is represented by a curved line on the graph, representing the set of real numbers. It is sometimes necessary to choose one particular graph based on the particular rational function f(x), when dealing with real number calculus or algebra.

The main advantages of geometric graph approaches to irrational functions are accuracy and speed. For finite values of the functions, geometric graph approaches are accurate because the area under the curve is definite. In geometric graph approaches, the area under the curve is called the limit of convergence, which is equivalent to the exact value of the function on the graph.

For complex functions, however, such as those representing complex numbers or complex tangent series, geometric approaches may not give the correct results. On the other hand, when the function has only zero slope on the horizontal, and if the horizontal axis has arithmetic values, then the analytical method would give the exact value of the function on the horizontal graph.

It is necessary to draw the graphs for every irrational number using the same scale, so that both the horizontal and vertical scales coincide. For example, a graph of a parabola on a plane can be plotted on a plane having one radii, whereas a graph of the parabola on a spherical surface can be plotted on a spherical plane having two radii. This means that the graphs must be similar for all the rimes of the complex number. The rimes have zero degrees of freedom, whereas the parabola graph has degrees of freedom.

A second example of graph the** behavior of rational functions i**ncludes the parabola and the hyperbola. Both the parabola and the hyperbola satisfy the analytic equation x = a sin (sin(x) * sin(z) + cos(z), where z is the angle of the tangent to the x-axis at rest. The analytic solution is the parabola’s mean value, which is the arithmetic mean of the solutions of the equations. In this example, both the parabola and the hyperbola Graphs with degrees of freedom are useful to obtain the mean values of the functions.

## Behaviors Describing the Behavior of Functions

In order to understand the behavior of functions, it is first necessary to grasp a bit more information regarding the symmetries of behavior that arise in the brain. If we know what the normal state of any particular function is, and how it changes as a function of some set of inputs, then it becomes fairly simple to see how behavior relates to the other components of the brain. Asymmetries in behavior of functions can be seen in a variety of functions.

For instance, in a speech there are tonal as well as morphological asymmetries, where different words produce different sounds or are processed in different ways, as a result of the fact that they come from different languages and cultures.

In the case of synchronicity, there are rhythmic and lyrical intervals in speech that enable it to have a certain level of sync with musical rhythms, which in turn enables speech to have a certain rhythm, a certain cadence, a certain range of pitches and so on. These things may not be consciously observable but they are there, and they are important to explain the behavior of functions. It will be useful however, for us to also understand the end behavior of functions: this involves describing the patterns of firing rates, spike activity and so on.

One way to describe the end behavior of functions is by noting that each function has its very own neighborhood of frequencies, strengths and dips. So, in our language for instance, the word “be”, has its own frequency and strength, and so does the word “thou” which has a completely different frequency and strength.

Then we have the word “beep” which has a very low frequency and is therefore a weak pulse, while the behavior of functions involving spoken sentences is such that it has high levels of accents, modulations, and so on. Thus, all the elements of speech, including the rhythm, accent, tempo, range of tones, and so on, have their own discrete domains of frequencies, and we can say that each behavior of functions has its corresponding frequency spectrum.