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:

\[Â yÂ =Â aÂ bÂ ^Â xÂ \]

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 ).**Â 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 $Â yÂ =Â fÂ (Â xÂ )Â $ consisting of the following values:

$Â yÂ =Â 125Â $ at $Â xÂ =Â 0Â $

$Â yÂ =Â 25Â $Â atÂ $Â xÂ =Â 1Â $

$Â yÂ =Â 5Â $ Â at Â $Â xÂ =Â 2Â $

$Â yÂ =Â 1Â $ or $Â xÂ =Â 3Â $

$Â yÂ =Â \frac { 1 } { 5 } Â $ at $Â xÂ =Â 4Â $

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.

\[Â yÂ =Â aÂ bÂ ^Â xÂ \]

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.

$Â yÂ =Â 81Â $ at $Â xÂ =Â 0Â $

$Â yÂ =Â 27Â $Â atÂ $Â xÂ =Â 1Â $

$Â yÂ =Â 9Â $ Â at Â $Â xÂ =Â 2Â $

$Â yÂ =Â 3Â $ or $Â xÂ =Â 3Â $

$Â yÂ =Â \frac { 1 } { 2 } Â $ or $Â xÂ =Â 4Â $

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**.*