Mathematical Investigation of Functions

Abstract

Generally, when the independent variable of a given exponential function is used as an exponent, the function is considered an exponential. Thus, the following can be examples of exponential functions: $f(x) = ab^x + c$, $f(x) = ae^bx + c$, or $f(x) = e^{a^2+bx+c}$. However, deriving functions of these types given the set of ordered pairs is difficult. This study was conducted to derive formulas for the arbitrary constants a ,b, and $c$ of the exponential function $f(x) = ab^x + c$. It applied the inductive method by using definitions of functions to derive the arbitrary constants from the patterns produced. The findings of the study were: a) For linear, given the table of ordered pairs, equal differences in $x$ produce equal first differences in $y$; b) for quadratic, given the table of ordered pairs, equal differences in $x$ produce equal second differences in $y$; and c) for an exponential function, given a table of ordered pairs, equal differences in $x$ produce a common ratio in the first differences in y. The study obtained the following forms: $b=\sqrt[d]{r}$, $a=\frac{q}{b^n {(b^d-1)}}$, $c=p-ab^n$. Since most models developed used the concept of linear and multiple regressions, it is recommended that pattern analysis be used specifically when data are expressed in terms of ordered pairs.