This question aims to find whether the given table consisting of **function f** represents **exponential growth** or not.

Exponential growth is also called a decay function when function is decreasing. A **decay function** is a type of function that decays with the factor of the number. When the function increases, it shows the growth of a given function, also called **exponential growth**. These functions are represented in the form of:

\[ y = a b ^ x \]

In the above formula, **a** represents the **initial value** of the function and **b** determines whether the function is **increasing** or **decreasing**. **For example,** if the value of b is **greater than two,** then it represents the growth of the function **f ( x ).** But when the value of b is **less than two,** then it means it is a **decay function** as the function is decreasing.

## Expert Answer

Consider a table of function $ y = f ( x ) $ consisting of the following values:

$ y = 125 $ at $ x = 0 $

$ y = 25 $ at $ x = 1 $

$ y = 5 $ at $ x = 2 $

$ y = 1 $ or $ x = 3 $

$ y = \frac { 1 } { 5 } $ at $ x = 4 $

The value of **x increases** by 1, which shows the decrease in the function **y = f ( x )** by the factor of **five**. It means the given function represents the exponential decay function.

## Numerical Solution

**The function y = f ( x ) is a decay function as it shows exponential decay.**

## Example

The function y = f ( x ) is given. Find whether the function is increasing or decreasing.

The function that is **increasing** shows **exponential growth** while the **decreasing function** shows exponential decay.

\[ y = a b ^ x \]

In the above formula, a represents the initial value of the function and b determines whether the function is increasing or decreasing. For example, if the value of b is **greater** than two, then it represents the growth of the function f ( x ). But when the value of b is **less than** two, then it means it is a decay function as the function is decreasing.

$ y = 81 $ at $ x = 0 $

$ y = 27 $ at $ x = 1 $

$ y = 9 $ at $ x = 2 $

$ y = 3 $ or $ x = 3 $

$ y = \frac { 1 } { 2 } $ or $ x = 4 $

The above function is decreasing with a factor of **3** as value of x is increasing, which confirms the decay function.

**The function y = f ( x ) is a decay function as it shows exponential decay.**

*Image/Mathematical drawings are created in Geogebra**.*