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Which table represents exponential growth.

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.

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