The exponential function is a type of mathematical function.
In order to know the meaning of the term exponential function that concerns us now, it is necessary to first discover the etymological origin of the two words that give it shape:
-Function, first of all, derives from Latin, exactly from “functio”, which can be translated as “exercise” or “function”. Likewise, it comes from the verb “fungi”, which is synonymous with “to fulfill” or “to perform a task”.
-Exponential, secondly, also derives from Latin. It means “growth that increases more and more rapidly” and is the result of the sum of several lexical components of said language: the prefix “ex-”, which is synonymous with “outwards”; the verb “ponere”, which can be translated as “put”; the particle “-nt-”, which is used to indicate agent, and the suffix “-al”, which means “relative to”.
A type of mathematical function
In the field of mathematics , a function is a link between two sets through which each element of the first set is assigned a single element of the second set or none. Exponential , on the other hand, is an adjective that qualifies the type of growth whose pace increases increasingly faster.
According to their characteristics, there are various types of mathematical functions . An exponential function is a function that is represented by the equation f(x) = aˣ , in which the independent variable ( x ) is an exponent.
Characteristics of the exponential function
An exponential function, therefore, allows us to refer to phenomena that grow increasingly rapidly . Take the case of the development of a bacterial population: a certain species of bacteria that, every hour, triples its number of members. This means that, every x hours, there will be 3ˣ bacteria .
An exponential function is reflected in the equation f(x) = aˣ
The exponential function indicates that, starting from a bacteria:
After one hour: f(1) = 3¹ = 3 (there will be three bacteria)
After two hours: f(2) = 3² = 9 (there will be nine bacteria)
After three hours: f(3) = 3³ = 27 (there will be twenty-seven bacteria)
Etc.
Returning to the equation f(x) = aˣ , we must take into account that a is the base , while x is the exponent. In the case of the example of bacteria that triple every hour, the base is 3 , while the exponent is the independent variable that changes over time.
In exponential functions, the set of real numbers constitutes their domain of definition. The function itself, on the other hand, is its derivative .
Other properties
In addition to everything stated above, we cannot ignore another series of relevant data about the exponential function such as the following:
-It is a continuous class.
-It is determined that it is increasing if a > 1 and that it is decreasing if a