If one were to construct a hierarchy from the various cognitive processes involved in the brain, the most logical order would be from left to right in that order rational thinking and linguistic behavior take place. It is interesting to note that such cognitive processes, while related in many ways, are in fact independent of one another. For instance the linguistic approach is associated with the left side of the brain and rational thinking with the right. It is interesting to note that the **end behavior of rational functions** is identical on both hemispheres. The left side is where rational thinking occurs and the right side where linguistic activity takes place.

One example of an **end behavior of rational functions** would be the graphing function, which is well known to be rationally based and so, on the basis of this alone, we could say that there is a right and a left side to every rational thought. This is also a great example of the cognitively independent aspects of reasoning. Another example is the so called “oblique” or “cross-sensitive” triangle. When we draw a line from left to right with a straight and a curved side going from top to bottom, this is a right to left horizontal line, the vertical axis of the triangle is along the left side of the horizontal line and the y-axis must point upwards. Hence it can be concluded that the horizontal line encloses a plane of symmetry and so it forms an oblique or cross-sectional representation.

Similarly, the vertical approach brings about an **end behavior of rational functions** on the basis of a mathematical equation. For instance the x is tangent gives the z value of the angle between the x-axis and the horizontal at right angles to the x-axis. It is therefore axial to the x-axis. The tangent of the plotted function would then be the y value of this angle. The horizontal asymptote then is equal to the x-intercept of the tangent.

The rational function f is called the rational function of a set of arbitrary variables A, B, C where A is one of the inputs in the calculation of B and C is any other variable. In case the function f(x) is a closed form then the integral formula is used to get the value of the function at the end of the interval x for each input A. It is then said to be the mean value of the function at the end of the interval x for all inputs A. So we see that the mean function is the end behavior of rational functions that is equal to the x-intercept of the tangent.

Graphs are used extensively in geometry, computer science and related applications. We may see the graphical expressions of a set of functions on the real axis, the x-axis being viewed at right angles to the plotted horizontal asymptote. The function f(x) plotted on a horizontal bar is termed as the point of intersection of two curves on the x-axis. So for every point on the graph, the corresponding equation is called an asymptote, or the curve on the x-axis can be graphed so that the set of points on the graph intersect at a point called the mean point on that graph.

For our purposes therefore we take the point of intersection of the tangent curves and the mean points on the graph as the end behavior of rational functions on a plane passing through the points on the graph. This plane is then seen as a straight line on the chart so that we may plot the functions f(x) on the corresponding horizontal arrow symbols. This means that the end behavior of rational functions can be written in the form of a plane passing through the points on the graph.

Thus, the term ‘asymmetrical triangles’ is used here to designate the curved lines on the chart. Graphically it might not be clearly visible how the slopes of the y-axis are related to the set of end behavior of rational functions, but the ‘asymmetrical’ concept is easily understood in this context. Finally I would like to say that the symmetrical triangles in mathematics are the symmetric equivalents of the parabola.

## Rational Function End Behavior

We know the end behavior of rational functions by use of a graphical representation. The end behavior of rational functions is represented by a function which satisfies all the graphs in existence. Let us use the prime function as an example. The prime function is the first power of a number, so it satisfies the graph of xy-axis when x = 1 and y = 0. Therefore, the prime function is called x-axis because it is at the origin. It can be viewed as a point on a graph which represents the infinitely plotted function.

The end behavior of rational functions will be affected by a few factors. These factors are called set ups. In order to plot the graph, one has to decide where and when to start the graph and then determine the range of the graph intercepts so that one ends up with a range of data that satisfies the equations equal to zero. Once this is decided, it is relatively easy to plot a good graph that meets all the set ups. The important point here is that the set ups and the location of the dots on the graph intersect at a particular point where the dot products equal infinity and therefore, infinity is a solution for all the equations.

The end behavior of rational functions is also influenced by a few symmetries. A symmetric approach implies that if one starts from any set up such as the x-axis starting at zero then each of the derivatives will be linearly scaled by some factor. This concept is used to determine the x-intercepts and hence, plot the end behavior of rational functions. In a graphical language, this is called a non-symmetrical y-axes and y-coils.