  The article presents mathematical generalization of results which originated
as solutions of practical problems, in particular, the modeling of transitional
processes in electrical circuits and problems of resource allocation. However,
the presented findings have broader meaning and can be used for approximation
of transitional and other processes in different areas of science and
technology. We present discovered properties of sums of polynomial, power, and
exponential functions of one variable. It is shown that for an equation
composed of one type of function there is a corresponding equation composed of
functions of the other type. The number of real solutions of such equations and
the number of characteristic points of certain appropriate corresponding
functions are closely related. In particular, we introduce a method similar to
Descartes Rule of Signs that allows finding the maximum number of real
solutions for the power equation and equation composed of sums of exponential
functions. The discovered properties of these functions allow us to improve the
adequacy of mathematical models of real phenomena.
