Comparison with Alternative Functions

Oct 13, 2025 by recoveryshake.com 12 views

Introduction

Park rangers released 4 fish into a pond in the year 0. Each year, the number of fish triples, resulting in a significant increase in the population. In this article, we will explore how to model this scenario using a mathematical function.

Understanding the Problem

The problem states that there are 4 fish in the pond initially. Each year, the number of fish triples. This means that if there are x fish in a particular year, there will be 3x fish in the next year. We need to find a function that represents this growth pattern.

Mathematical Modeling

To model this scenario, we can use a function of the form f(x) = a * b^x, where a is the initial value and b is the growth factor. In this case, the initial value is 4, and the growth factor is 3.

Function Representation

The function that represents this scenario is f(x) = 3(4)^x. This function takes an integer x as input and returns the number of fish in the pond after x years.

Explanation

Let's break down the function f(x) = 3(4)^x:

The term (4)^x represents the growth of the fish population. Since the population triples each year, we can represent this growth using the exponential function 4^x.
The coefficient 3 is the initial value of the population, which is 4 fish.
The function f(x) = 3(4)^x represents the population after x years.

Example Calculations

To illustrate how this function works, let's calculate the population for a few years:

Year 0: f(0) = 3(4)^0 = 3(1) = 3
Year 1: f(1) = 3(4)^1 = 3(4) = 12
Year 2: f(2) = 3(4)^2 = 3(16) = 48
Year 3: f(3) = 3(4)^3 = 3(64) = 192

As we can see, the population triples each year, resulting in a significant increase in the number of fish.

Comparison with Alternative Functions

One alternative function that might seem plausible is f(x) = 3(x)^4. However, this function does not accurately represent the growth pattern described in the problem.

The term (x)^4 represents a growth pattern where the population increases by a factor of x^4 each year. This is not consistent with the problem statement, which states that the population triples each year.
The function f(x) = 3(x)^4 would result in a much faster growth rate than the actual population growth.

Conclusion

In conclusion, the function f(x) = 3(4)^x accurately represents the growth pattern of the fish population in the pond. This function takes into account the initial value of the population and the growth factor, resulting in a realistic model of the population growth.

Code Implementation

Here is a Python code snippet that implements the function f(x) = 3(4)^x:

def fish_population(x):
    """
    Calculate the fish population after x years.

    Args:
        x (int): The number of years.

    Returns:
        int: The fish population after x years.
    """
    return 3 * (4 ** x)

You can use this function to calculate the population for a given number of years. For example:

print(fish_population(0))  # Output: 3
print(fish_population(1))  # Output: 12
print(fish_population(2))  # Output: 48
print(fish_population(3))  # Output: 192

Note that this function assumes that the input x is a non-negative integer. If you need to handle negative or non-integer inputs, you will need to modify the function accordingly.