# Exponential Function Examples

The **exponential function** is (as its name implies), a **function that is personified by means of the «equation f (x) = aˣ»,** this is characterized because the variable “x” is known as an exponent.

In this article, we will give you the functions that the exponential function has in mathematics and also the best examples about it.

## Exponential Function Functions

The exponential function helps us to manifest phenomena that increase rapidly. An example of this is the development of a bacterium in the population, since if it is infectious, every certain period of time it will triple its number of components. This indicates that every certain period of time, there will be 3ˣ bacteria.

These indicate that

- After the first hour: f (1) = 3
^{1}= 3. You will have three more bacteria. - After two hours: f (2) = 3
^{2}= 9. You will have nine more bacteria. - After three hours: f (3) = 3³ = 27. There will be twelve bacteria.

And so on.

Returning to the **equation f (x) = aˣ** , we must mention that the base is “a” and “x” is known as the exponent.

In the example mentioned above, on the triplication of the bacteria according to the elapsed time, the base of the exercise is the number “3”, and the exponent is the independent variation that is modified in the period of time.

These are constituted by their definition. However, they are derived from their own function. It is also important to mention that the exponential function is continuous. This is classified as increasing if a> 1 and as decreasing if a <1.

These can be used in a myriad of sectors that are performed to solve a large number of calculations. The exponential function is used precisely and definitively in work situations with population increases in a given place; at the level of added interests and in reference to the economic situation and in turn, it is used for work with the well-known radiant loss.

## Exponential Function Examples

Next, we will give you the best **examples of an exponential function** so that you can learn to develop it in the correct way.

- 67-X = 62X + 1

Now we will explain how to solve this exercise step by step:

**67-x = 62x + 1**

**7 – x = 2x + 1**

**7 – 1 = 2x + x**

**6 = 3x**

**x = 2**

- 9 (x2) = 33x + 2

The steps to solve this exercise are:

**9x + 2 = 35x-8**

**(32) x + 2 = 35x-8**

**32x + 4 = 35x-8**

**2x + 4 = 5x – 8**

**4 + 8 = 5x – 2x**

**12 = 3x**

**x = 4**

- (1/2) 6-x = 2

The resolution procedure for this exercise is:

**(1/2) 6-x = 2**

**(2-1) 6-x = 2**

**2–6 + x = 2**

**-6 + x = 1**

**x = 7**

- f (x) = ax

We will explain this example to you using the following graph:

- f (x) = -ax

As you can see through the following image, this exercise is solved as follows:

- f (x) = 3ax

This graph explains the result of the proposed example in a simple way:

- f (x) = (2/5) x

In this example we will provide you with the solution through the following image:

- f (x) = 8 (4) -x – 2

Finally, you can observe in this last exercise, the correct way to solve this exponential function exercise.

Throughout this article, you were able to learn about the definition, functions and the correct way to **use and solve the exponential function,** not to mention that we were able to reach the conclusion about the importance of knowing this mathematical property, since it influences a lot in the **ease of solving mathematical exercises.**