In order to know the meaning of the term exponential function that now concerns us, it is necessary first to discover the etymological origin of the two words that give it shape:

-Function, first of all, it derives from Latin, exactly from “functio”, which can be translated as “exercise” or “function”. Likewise, that comes from the verb “fungi”, which is synonymous with “fulfill” or “perform a task”.

-Exponential, secondly, it also derives from Latin. It means “growth that increases more and more rapidly” and is the result of the sum of various lexical components of that language: the prefix “ex-“, which is synonymous with “outward”; the verb “put”, 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 **math**, a **function** it is a link between two sets by which each element of the first set is assigned a single element of the second set or none at all. **Exponential**, on the other hand, it is an adjective that describes the type of growth whose rate is increasing faster and faster.

According to their characteristics, there are different types of **math 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 faster and faster**. Take the case of the development of a bacterial population: a certain species of **bacterium** that, every hour, **triples** its number of members. This means that each **x** hours, there will be **3ˣ bacteria**.

The exponential function indicates that, starting from a bacterium:

After an hour: **f (1) = 3¹ = 3** (there will be three bacteria)

Two hours later: **f (2) = 3² = 9** (there will be nine bacteria)

After three hours: **f (3) = 3³ = 27** (there will be twenty-seven bacteria)

Etc.

Picking up the equation **f (x) = aˣ**, keep in mind that a is the **base**, while **x** is the exponent. For the example of bacteria that triples 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 all the above, we cannot ignore another series of relevant data on the exponential function such as the following:

-It’s a continuous class.

-It is determined that it is increasing if a> 1 and that it is decreasing if a