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.
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