Calculate Function Values: F(x) = 5^(x-3)

Let's dive into calculating function values! In this article, we'll explore the function f(x) = 5^(x-3) and determine the outputs for specific inputs. We will focus on finding the values of f(6) and f(8). Understanding how to evaluate functions is a fundamental concept in mathematics, and this example will provide a clear, step-by-step guide. So, let's get started and unravel the mystery of this exponential function.

Understanding the Function f(x) = 5^(x-3)

Before we jump into calculations, let's break down the function f(x) = 5^(x-3). This is an exponential function, where the variable x appears in the exponent. The base of the exponent is 5, and we are subtracting 3 from the input x before using it as the power. Exponential functions are powerful tools in mathematics and are used to model various phenomena like population growth, radioactive decay, and compound interest.

The key here is to understand how the input x affects the output f(x). When we substitute a specific value for x, we first subtract 3 from it, and then we raise 5 to that power. This simple process is the core of evaluating any function, and mastering it will unlock a deeper understanding of mathematical concepts. Understanding exponential functions like this is crucial for various applications in science and engineering. The shape of the graph and the rate of change are distinctive features that make them indispensable in mathematical modeling. Consider exploring further resources to solidify your understanding of exponential functions and their diverse applications. This foundational knowledge will help you not only solve the problem at hand but also tackle more complex mathematical challenges in the future.

Calculating f(6)

Now, let's calculate the value of f(6). To do this, we substitute x = 6 into the function f(x) = 5^(x-3). This gives us f(6) = 5^(6-3). The next step is to simplify the exponent. We have 6 - 3 = 3, so the expression becomes f(6) = 5^3. Finally, we evaluate 5 cubed, which is 5 * 5 * 5 = 125. Therefore, f(6) = 125.

This calculation demonstrates the straightforward process of evaluating a function at a specific point. We replace the variable with the given value, simplify the expression, and arrive at the output. This process is fundamental to understanding functions and their behavior. Mastering function evaluation is a cornerstone of mathematical proficiency. It enables us to predict outputs for different inputs and to analyze the relationships between variables. Furthermore, this skill is not limited to mathematics; it is widely applicable in various fields, including computer science, engineering, and finance. Think about how spreadsheets work – they essentially perform function evaluations based on your formulas and input data. So, by understanding this process, you are not just learning math; you are developing a valuable analytical skill that will serve you well in many areas of life. Practice this process with different functions and values to reinforce your understanding and build confidence in your ability to tackle mathematical problems.

Calculating f(8)

Next, we'll calculate the value of f(8). We follow the same process as before, substituting x = 8 into the function f(x) = 5^(x-3). This yields f(8) = 5^(8-3). Simplifying the exponent, we find 8 - 3 = 5, so the expression becomes f(8) = 5^5. Now, we need to evaluate 5 to the power of 5, which means 5 * 5 * 5 * 5 * 5. This equals 3125. Thus, f(8) = 3125.

This calculation further illustrates the power of exponential growth. Notice how the function value increases significantly as the input increases. This is a characteristic of exponential functions, where even small changes in the input can lead to substantial changes in the output. Understanding exponential growth is crucial in many real-world applications, from predicting population trends to analyzing financial investments. For example, the spread of a virus often follows an exponential pattern, as does the growth of money in a compound interest account. By understanding how these functions behave, we can make informed decisions and predictions about the world around us. Consider exploring real-world examples of exponential growth and decay to deepen your understanding of this important mathematical concept. This will not only enhance your mathematical skills but also provide valuable insights into various aspects of science, economics, and everyday life. The ability to recognize and analyze exponential patterns is a valuable asset in today's data-driven world.

Summary of Results

In summary, we have calculated the function values for f(6) and f(8). We found that f(6) = 125 and f(8) = 3125. These calculations demonstrate the process of evaluating an exponential function at specific points. By substituting the input value into the function and simplifying the expression, we can determine the corresponding output value.

This exercise highlights the fundamental nature of function evaluation in mathematics. It is a skill that is used across various mathematical disciplines and is essential for understanding the behavior of functions. Reviewing these calculations and the underlying principles will strengthen your understanding of functions and their applications. Remember, practice is key to mastering mathematical concepts. Try evaluating the function at other points, such as f(4) or f(2), to further solidify your understanding. You can also explore other exponential functions with different bases and exponents to broaden your knowledge. The more you practice, the more comfortable and confident you will become in working with functions. Function evaluation is not just a mathematical exercise; it is a tool for understanding and modeling the world around us. By mastering this skill, you are equipping yourself with a powerful analytical tool that will serve you well in many academic and professional pursuits.

Conclusion

We have successfully calculated the function values f(6) = 125 and f(8) = 3125 for the function f(x) = 5^(x-3). This exercise reinforces the fundamental process of function evaluation and highlights the characteristics of exponential functions. Remember, consistent practice is crucial for mastering mathematical concepts. Keep exploring different functions and their properties to deepen your understanding. Understanding these concepts is not just about getting the right answer; it's about developing a way of thinking that can be applied to a wide range of problems. Mathematics is a powerful tool, and by understanding its core principles, you unlock the ability to solve complex problems and make informed decisions. Don't be afraid to challenge yourself with more complex problems, and always remember that the journey of learning mathematics is a continuous one. Embrace the challenges, celebrate your successes, and keep exploring the fascinating world of mathematics!

For further learning on exponential functions, consider visiting Khan Academy's Exponential Functions Page.