Unveiling Exponential Functions: Solving For F(x)

by Alex Johnson 50 views

Welcome, math enthusiasts! Today, we're diving into the fascinating world of exponential functions. Specifically, we'll learn how to complete the table of values for a given exponential function. Our guide for this exploration is the function f(x) = (1/3)^x. This journey will not only solidify your understanding of exponential functions but also equip you with the skills to confidently tackle similar problems. Get ready to flex your mathematical muscles, as we uncover the secrets of this intriguing function!

Decoding Exponential Functions: The Basics

Before we jump into calculations, let's brush up on the fundamentals of exponential functions. An exponential function is a function of the form f(x) = a^x, where 'a' is a positive constant (the base) and 'x' is the variable exponent. The base determines the growth or decay of the function. If 'a' is greater than 1, the function exhibits exponential growth; if 'a' is between 0 and 1, we observe exponential decay. In our case, f(x) = (1/3)^x, the base is 1/3, which is less than 1. This tells us that our function will experience exponential decay – the values of f(x) will decrease as x increases.

Understanding the components of the exponential function is crucial before filling the table. The variable 'x' represents the input value, and f(x) or y represents the output value. When we plug in different values of x into the function, we'll get corresponding values for f(x). We must understand the concept of exponentiation. Exponentiation is a mathematical operation, written as a^n, involving two numbers, the base a and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base. For example, 3^4 = 3 * 3 * 3 * 3 = 81. In our function, we are using fractional values and negative values for x, so understanding the properties of exponents is necessary. The key is understanding that negative exponents involve reciprocals. a^(-n) = 1/a^n.

So, as we proceed to complete the table, let us remember these principles. The goal is to accurately calculate the f(x) values for the given x values. We'll be using this fundamental knowledge as we embark on our mission to complete the table of values. Remember, practice makes perfect, and with each calculation, we sharpen our skills and deepen our appreciation for these powerful functions!

Solving for the Table Values: Step-by-Step

Now, let's get down to the exciting part: completing the table! We have the function f(x) = (1/3)^x and need to find the values for f(x) when x is -2, -1, and 0. The table already provides values for x = 1 and x = 2. Let's tackle each missing value one at a time. This step-by-step approach ensures clarity and helps reinforce the underlying concepts.

  1. When x = -2: To find f(-2), we substitute -2 for x in the function: f(-2) = (1/3)^(-2). Remember our exponent rules? A negative exponent means we take the reciprocal of the base and raise it to the positive value of the exponent. So, (1/3)^(-2) becomes (3/1)^2, which simplifies to 3^2. Therefore, f(-2) = 9. We have successfully calculated the value for x = -2. So, we'll fill 9 in the table.

  2. When x = -1: Now, let's find f(-1). Substituting -1 for x: f(-1) = (1/3)^(-1). Again, the negative exponent means we take the reciprocal: (1/3)^(-1) becomes (3/1)^1, which simplifies to 3^1. Thus, f(-1) = 3. We have successfully calculated the value for x = -1. So, we'll fill 3 in the table.

  3. When x = 0: Let's find f(0). Plugging in 0 for x: f(0) = (1/3)^0. Any non-zero number raised to the power of 0 equals 1. Therefore, f(0) = 1. We have successfully calculated the value for x = 0. So, we'll fill 1 in the table.

By following these steps, we've filled the missing values in our table. With each calculation, we are gaining a better understanding of how exponential functions work. We must understand the properties of the exponents. Because of our thorough understanding of exponent rules, completing this table has been a breeze. Understanding the role of the base is also important. As we continue to delve deeper into these functions, we are building a solid foundation in the mathematical world.

The Completed Table: A Summary

Now that we've calculated all the missing values, let's present the completed table. This summary offers a clear view of the relationship between x and f(x) for our exponential function. The purpose of presenting the finished table is to enable easy recognition and understanding of the function's output for various input values. This table is a visual testament to the function's behavior.

x f(x)
-2 9
-1 3
0 1
1 1/3
2 1/9

As we can see, as the x values increase, the f(x) values decrease, which aligns with our earlier understanding of exponential decay. The table is a perfect illustration of how the function behaves. Understanding this relationship is a key aspect of mastering exponential functions. Visualizing the relationship between inputs and outputs allows for an intuitive understanding. It becomes a tool for making predictions and solving real-world problems. We've conquered the challenge and successfully completed the table of values! The values are in perfect accordance with the function rule and properties of exponents, demonstrating our full understanding of this concept. Keep in mind that practice with multiple examples can strengthen this skill. With our newfound mastery, we are ready to tackle many mathematical problems.

Conclusion: Embracing the Power of Exponential Functions

Congratulations! You've successfully completed the table of values for an exponential function. We started with a basic exponential function, f(x) = (1/3)^x, and through step-by-step calculations, we were able to find the corresponding f(x) values for the given x values. We've explored the properties of exponents, especially those involving negative exponents and the exponent 0, and we've seen how the function demonstrates exponential decay. This is a very valuable skill, and we can now build upon this knowledge. Remember, the key to success is understanding the underlying principles and practicing regularly. We've not only completed the table but also deepened our understanding of exponential functions. This understanding is useful in many real-world applications.

Exponential functions are fundamental in various fields, including science, finance, and engineering. From modeling population growth and radioactive decay to calculating compound interest, these functions are essential tools. By mastering these functions, we gain the ability to analyze and predict complex phenomena. So, embrace the power of exponential functions, keep practicing, and explore the fascinating world of mathematics!

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