Evaluating the Population Function

Oct 13, 2025 by miodeya.com 46 views

Introduction

Exponential functions are a fundamental concept in mathematics, used to model various real-world phenomena, including population growth, chemical reactions, and financial investments. In this article, we will delve into the world of exponential functions, focusing on a specific case study: the population of a community modeled by the function $p(x) = 2,400(1.025)^x$ , where $x$ represents the number of years since the population started being recorded.

What is an Exponential Function?

An exponential function is a mathematical function of the form $f(x) = ab^x$ , where $a$ and $b$ are constants, and $x$ is the variable. The base $b$ is a positive number, and the exponent $x$ is a real number. Exponential functions have the following properties:

Growth or Decay: Exponential functions can model both growth and decay phenomena. If $b > 1$ , the function represents growth, while if $0 < b < 1$ , the function represents decay.
Multiplicative: Exponential functions are multiplicative, meaning that the output of the function is obtained by multiplying the input by a fixed constant.
Continuous: Exponential functions are continuous, meaning that the function can be evaluated at any point in its domain.

The Population Function

The population function $p(x) = 2,400(1.025)^x$ represents the population of a community at any given time $x$ . The function has the following characteristics:

Initial Population: The initial population is $p(0) = 2,400(1.025)^0 = 2,400$ .
Growth Rate: The growth rate of the population is $1.025$ , which means that the population increases by $2.5\%$ each year.
Population at Time $x$ : The population at time $x$ is given by the function $p(x) = 2,400(1.025)^x$ .

Evaluating the Population Function

To evaluate the population function at a specific time $x$ , we can use the following steps:

Plug in the value of $x$ : Substitute the value of $x$ into the function $p(x) = 2,400(1.025)^x$ .
Calculate the result: Use a calculator or computer software to evaluate the function at the given value of $x$ .

Example: Evaluating the Population Function at $x = 10$

To evaluate the population function at $x = 10$ , we can follow the steps outlined above:

Plug in the value of $x$ : Substitute $x = 10$ into the function $p(x) = 2,400(1.025)^x$ .
Calculate the result: Using a calculator or computer software, we get $p(10) = 2,400(1.025)^{10} \approx 3,517.19$ .

Interpreting the Results

The result $p(10) = 3,517.19$ means that the population of the community at time $x = 10$ is approximately $3,517.19$ .

Conclusion

In this article, we have explored the concept of exponential functions and their application in modeling population growth. We have used the population function $p(x) = 2,400(1.025)^x$ to demonstrate the properties of exponential functions and how to evaluate them at specific times. By understanding exponential functions, we can better model and analyze real-world phenomena, including population growth, chemical reactions, and financial investments.

Glossary

Exponential Function: A mathematical function of the form $f(x) = ab^x$ , where $a$ and $b$ are constants, and $x$ is the variable.
Growth Rate: The rate at which a quantity increases over time.
Population Function: A mathematical function that models the population of a community over time.
Exponent: A real number that is raised to a power in an exponential function.
Base: A positive number that is raised to a power in an exponential function.

Introduction

Exponential functions are a fundamental concept in mathematics, used to model various real-world phenomena, including population growth, chemical reactions, and financial investments. In this article, we will provide a comprehensive Q&A guide to help you understand exponential functions and their applications.

Q1: What is an Exponential Function?

A: An exponential function is a mathematical function of the form $f(x) = ab^x$ , where $a$ and $b$ are constants, and $x$ is the variable. The base $b$ is a positive number, and the exponent $x$ is a real number.

Q2: What are the Properties of Exponential Functions?

A: Exponential functions have the following properties:

Growth or Decay: Exponential functions can model both growth and decay phenomena. If $b > 1$ , the function represents growth, while if $0 < b < 1$ , the function represents decay.
Multiplicative: Exponential functions are multiplicative, meaning that the output of the function is obtained by multiplying the input by a fixed constant.
Continuous: Exponential functions are continuous, meaning that the function can be evaluated at any point in its domain.

Q3: How Do I Evaluate an Exponential Function?

A: To evaluate an exponential function, you can use the following steps:

Plug in the value of $x$ : Substitute the value of $x$ into the function $f(x) = ab^x$ .
Calculate the result: Use a calculator or computer software to evaluate the function at the given value of $x$ .

Q4: What is the Initial Population of a Community Modeled by an Exponential Function?

A: The initial population of a community modeled by an exponential function is given by the function $p(0) = a$ , where $a$ is the initial population.

Q5: How Do I Find the Growth Rate of a Population Modeled by an Exponential Function?

A: To find the growth rate of a population modeled by an exponential function, you can use the following formula:

Growth Rate: $r = \frac{b - 1}{b}$ , where $b$ is the base of the exponential function.

Q6: What is the Population at Time $x$ of a Community Modeled by an Exponential Function?

A: The population at time $x$ of a community modeled by an exponential function is given by the function $p(x) = a(b)^x$ , where $a$ is the initial population and $b$ is the growth rate.

Q7: How Do I Use Exponential Functions to Model Real-World Phenomena?

A: Exponential functions can be used to model various real-world phenomena, including:

Population Growth: Exponential functions can be used to model population growth, where the population increases or decreases over time.
Chemical Reactions: Exponential functions can be used to model chemical reactions, where the concentration of a substance increases or decreases over time.
Financial Investments: Exponential functions can be used to model financial investments, where the value of an investment increases or decreases over time.

Q8: What are Some Common Applications of Exponential Functions?

A: Exponential functions have many common applications, including:

Biology: Exponential functions are used to model population growth, chemical reactions, and other biological phenomena.
Economics: Exponential functions are used to model financial investments, inflation, and other economic phenomena.
Physics: Exponential functions are used to model radioactive decay, chemical reactions, and other physical phenomena.

Conclusion

In this article, we have provided a comprehensive Q&A guide to help you understand exponential functions and their applications. Exponential functions are a fundamental concept in mathematics, used to model various real-world phenomena. By understanding exponential functions, you can better model and analyze real-world phenomena, including population growth, chemical reactions, and financial investments.

Glossary

Exponential Function: A mathematical function of the form $f(x) = ab^x$ , where $a$ and $b$ are constants, and $x$ is the variable.
Growth Rate: The rate at which a quantity increases over time.
Population Function: A mathematical function that models the population of a community over time.
Exponent: A real number that is raised to a power in an exponential function.
Base: A positive number that is raised to a power in an exponential function.