When analyzing the function (f(x) 3 left(frac{2}{5}right)^x), it's essential to understand the behavior of exponential functions based on the base they contain. This article will delve into why (f(x) 3 left(frac{2}{5}right)^x) is an example of exponential decay and provide a detailed explanation supported by examples and graphs.

Understanding Exponential Decay

Exponential functions are categorized based on the base of the exponent. If the base is between 0 and 1 ((0 ), the function exhibits exponential decay. Conversely, if the base is greater than 1 ((b > 1)), the function shows exponential growth.

Defining Exponential Decay

In the given function (f(x) 3 left(frac{2}{5}right)^x):

The coefficient 3 is a constant multiplier and does not affect whether the function is growing or decaying. The base of the exponent is (frac{2}{5}), which is a fraction less than 1 (0 (frac{2}{5}) 1).

When the base of an exponential function is between 0 and 1, as (x) increases, the value of the function (f(x)) decreases, leading to exponential decay.

Graphical Representation of Decay

To further illustrate this, let's consider the graph of (f(x) 3 left(frac{2}{5}right)^x). As shown below, the graph starts high and decreases towards zero as (x) increases.

The graph demonstrates that as (x) increases, the value of (f(x)) decreases, confirming that (f(x) 3 left(frac{2}{5}right)^x) represents exponential decay.

General Rule for Exponential Functions

Given the general rule, if the base of an exponential function is greater than 1, the function shows growth. For instance, an exponential function (g(x) b^x) with (b > 1) will exhibit growth. However, if the base is less than 1 but greater than 0, the function will exhibit decay. This aligns with the behavior of the function (f(x) 3 left(frac{2}{5}right)^x).

Multiplying by a Constant

Another important point is that multiplying an exponential function by a constant does not change whether it is growth or decay. The coefficient 3 in (f(x) 3 left(frac{2}{5}right)^x) simply scales the function vertically, but it does not alter its fundamental decay or growth nature.

Critical Analysis and Contours

Let's consider the critical behavior of the function at any value of (x). As (x) increases, the function (f(x) 3 left(frac{2}{5}right)^x) decreases. This behavior can be observed in the table and graph below:

The table shows that as (x) increases, the value of (f(x)) decreases, confirming the exponential decay behavior.

Additionally, if you mean to interpret the function in a specific context where the decay is particularly critical, the behavior is still consistent with decay. Understanding the behavior of each component of the function can provide deeper insights into its nature.

In summary, the function (f(x) 3 left(frac{2}{5}right)^x) is an example of exponential decay because the base of the exponent is a fraction between 0 and 1. This decay is further substantiated by both graphical and tabular representations of the function.