Which Equation Represents An Exponential Function That Passes Through The Point $(2,36)$?A. $f(x) = 4(3)^x$ B. $f(x) = 4(x)^{3 \cdot}$ C. $ F ( X ) = 6 ( 3 ) X F(x) = 6(3)^x F ( X ) = 6 ( 3 ) X [/tex] D. $f(x) = 6(x)^3$

Mar 11, 2025 by ADMIN 231 views

Exponential Functions: Understanding the Basics and Identifying the Correct Equation

Introduction

Exponential functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including science, engineering, and economics. An exponential function is a mathematical function that describes a relationship between two quantities, where one quantity is a constant power of the other. In this article, we will explore the concept of exponential functions, understand the characteristics of an exponential function, and identify the correct equation that represents an exponential function that passes through the point (2,36).

What is an Exponential Function?

An exponential function is a function that can be written in 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. The function f(x) = ab^x is an exponential function if and only if b is a positive number.

Characteristics of an Exponential Function

Exponential functions have several characteristics that distinguish them from other types of functions. Some of the key characteristics of an exponential function include:

Growth or Decay: Exponential functions can exhibit either growth or decay behavior, depending on the value of the base b. If b > 1, the function exhibits growth behavior, while if 0 < b < 1, the function exhibits decay behavior.
Asymptotes: Exponential functions can have horizontal asymptotes, which are horizontal lines that the function approaches as x approaches infinity or negative infinity.
Domain and Range: The domain of an exponential function is all real numbers, while the range is all positive real numbers.

Identifying the Correct Equation

To identify the correct equation that represents an exponential function that passes through the point (2,36), we need to analyze each of the given options.

Option A: f(x) = 4(3)^x

This equation represents an exponential function with a base of 3 and a coefficient of 4. To determine if this equation passes through the point (2,36), we can substitute x = 2 and f(x) = 36 into the equation.

f(2) = 4(3)^2 f(2) = 4(9) f(2) = 36

Since the equation f(x) = 4(3)^x passes through the point (2,36), this is a possible solution.

Option B: f(x) = 4(x)^3

This equation represents a polynomial function, not an exponential function. Therefore, it does not meet the criteria for an exponential function.

Option C: f(x) = 6(3)^x

This equation represents an exponential function with a base of 3 and a coefficient of 6. To determine if this equation passes through the point (2,36), we can substitute x = 2 and f(x) = 36 into the equation.

f(2) = 6(3)^2 f(2) = 6(9) f(2) = 54

Since the equation f(x) = 6(3)^x does not pass through the point (2,36), this is not a possible solution.

Option D: f(x) = 6(x)^3

This equation represents a polynomial function, not an exponential function. Therefore, it does not meet the criteria for an exponential function.

Conclusion

Based on the analysis of each of the given options, the correct equation that represents an exponential function that passes through the point (2,36) is f(x) = 4(3)^x. This equation meets the criteria for an exponential function and passes through the given point.

Exponential Functions in Real-World Applications

Exponential functions have numerous real-world applications in various fields, including science, engineering, and economics. Some examples of exponential functions in real-world applications include:

Population Growth: Exponential functions can be used to model population growth, where the population grows at a constant rate.
Radioactive Decay: Exponential functions can be used to model radioactive decay, where the amount of radioactive material decreases at a constant rate.
Financial Calculations: Exponential functions can be used to calculate compound interest, where the interest is compounded at a constant rate.

Common Mistakes to Avoid

When working with exponential functions, there are several common mistakes to avoid, including:

Confusing Exponential and Polynomial Functions: Exponential functions and polynomial functions are two distinct types of functions, and it is essential to understand the difference between them.
Not Checking the Domain and Range: Exponential functions have a specific domain and range, and it is essential to check these values to ensure that the function is well-defined.
Not Using the Correct Base: The base of an exponential function is a critical component, and it is essential to use the correct base to ensure that the function is well-defined.

Conclusion

Exponential functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including science, engineering, and economics. In this article, we have explored the concept of exponential functions, understood the characteristics of an exponential function, and identified the correct equation that represents an exponential function that passes through the point (2,36). We have also discussed the real-world applications of exponential functions and common mistakes to avoid when working with these functions.
Exponential Functions: A Q&A Guide

Introduction

Q1: What is an exponential function?

A1: An exponential function is a mathematical function that describes a relationship between two quantities, where one quantity is a constant power of the other. It can be written in the form f(x) = ab^x, where a and b are constants, and x is the variable.

Q2: What are the characteristics of an exponential function?

A2: Exponential functions have several characteristics that distinguish them from other types of functions. Some of the key characteristics of an exponential function include:

Growth or Decay: Exponential functions can exhibit either growth or decay behavior, depending on the value of the base b.
Asymptotes: Exponential functions can have horizontal asymptotes, which are horizontal lines that the function approaches as x approaches infinity or negative infinity.
Domain and Range: The domain of an exponential function is all real numbers, while the range is all positive real numbers.

Q3: How do I determine if a function is exponential?

A3: To determine if a function is exponential, you need to check if it can be written in the form f(x) = ab^x, where a and b are constants, and x is the variable. If the function can be written in this form, then it is an exponential function.

Q4: What are some common mistakes to avoid when working with exponential functions?

A4: Some common mistakes to avoid when working with exponential functions include:

Confusing Exponential and Polynomial Functions: Exponential functions and polynomial functions are two distinct types of functions, and it is essential to understand the difference between them.
Not Checking the Domain and Range: Exponential functions have a specific domain and range, and it is essential to check these values to ensure that the function is well-defined.
Not Using the Correct Base: The base of an exponential function is a critical component, and it is essential to use the correct base to ensure that the function is well-defined.

Q5: How do I graph an exponential function?

A5: To graph an exponential function, you need to use a graphing calculator or a computer program. You can also use a table of values to create a graph. The graph of an exponential function will have a characteristic "S" shape, with the function increasing or decreasing rapidly as x approaches infinity or negative infinity.

Q6: What are some real-world applications of exponential functions?

A6: Exponential functions have numerous real-world applications in various fields, including science, engineering, and economics. Some examples of exponential functions in real-world applications include:

Population Growth: Exponential functions can be used to model population growth, where the population grows at a constant rate.
Radioactive Decay: Exponential functions can be used to model radioactive decay, where the amount of radioactive material decreases at a constant rate.
Financial Calculations: Exponential functions can be used to calculate compound interest, where the interest is compounded at a constant rate.

Q7: How do I solve exponential equations?

A7: To solve exponential equations, you need to use the properties of exponents. You can use the following properties to solve exponential equations:

Product of Powers Property: a^m * a^n = a^(m+n)
Power of a Power Property: (a^m)n = a^(m*n)
Quotient of Powers Property: a^m / a^n = a^(m-n)

Q8: What are some common exponential functions?

A8: Some common exponential functions include:

f(x) = 2^x: This function is an exponential function with a base of 2.
f(x) = 3^x: This function is an exponential function with a base of 3.
f(x) = 4^x: This function is an exponential function with a base of 4.

Conclusion

Exponential functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including science, engineering, and economics. In this article, we have provided a Q&A guide to help you understand exponential functions better. We have covered topics such as the characteristics of exponential functions, how to determine if a function is exponential, and how to solve exponential equations. We have also discussed some common mistakes to avoid when working with exponential functions and some real-world applications of exponential functions.