Introduction
The random house dictionary defines a function as a factor dependent on other factors; for example, a price is a function of supply and demand in microeconomics. A function is therefore made of two sets of objects and a correspondence that associates the object in one of the sets with another set, For example, each real number having exactly ones square and each price of an item relatable to the supply and demand of the said item. A function is also defined as a single-valued mapping from one set that is the domain to another that is the range.
Functions are mainly applicable in every aspect of applied mathematics. Functions apply to statistician using probability function to predict the occurrence of an event, a climatologist predicting the median temperature and rainfall of a geographical area etcetera.
The Concept function
The evolution of the concept function dates back to 4,000 years ago, and the idea was evolving to close to 300 years in connection to calculus and analysis. Functions evolved from the attempt to visualize geometric and algebraic formulas which later changed to logical and algebraic conception. Calculus, as it’s known today, did not develop in functions form rather Newton and Leibniz introduced calculus in the form of geometric curves. Isaac Newton further research into algebraic formulas led to him applying the power series, the roles of the symbols that arose in the equation and independent to the original curve.
Beginning of the 18th century was the separation of the geometric objects and the introduction function as an algebraic formula. In 1748 Euler attempted to study the concepts in analytical geometry that would enable the study of calculus. Euler began by asserting his claim that mathematical analysis was a science of variables and functions. A function of a variable quantity is an analytical expression composed in any manner from the variable quantity and numbers of constant quantity (Euler) was Euler’s first definition of the term function.
The development of tools of Calculus was also instrumental in the development of the function concept. Amongst the devices was the vibrating string problem by d’Alembert that arose from the “article of faith” that stated that if two analytic expressions agree on an interval, they agree everywhere. The ideology was that an independent variable in analytic expression ranges over the entire domain of real numbers without restriction.
Fourier contribution between 1768 and 1830 arose from the problem of heat flow in materials. Fourier considered temperature to be a function of space and time, and there existing a possibility of developing a trigonometric series function in a suitable interval. The probem was taken up in 1805 by Dirichlet who developed the Fourier series casting the definition of a function as a correspondence between two variables where any value of an independent variable could be associated by one value of the dependent variable and was the basis through which the function concept developed from.