Evaluating Exponential Functions: G(x) = 8^x At X = -2

Understanding Exponential Functions

In this comprehensive guide, we will delve into the evaluation of the exponential function g(x) = 8^x at the specific value of x = -2. Exponential functions are fundamental in mathematics and appear in various real-world applications, including finance, physics, and computer science. Understanding how to evaluate these functions is crucial for solving complex problems and making informed decisions. We'll break down the process step-by-step, ensuring you grasp the underlying concepts and can apply them effectively. This exploration will not only focus on the calculation but also on the conceptual understanding of what it means to raise a number to a negative power. By the end of this discussion, you will be well-equipped to tackle similar problems and appreciate the elegance and power of exponential functions. Let’s embark on this mathematical journey together, unraveling the intricacies of exponential functions and their evaluations.

Before we dive into the specific evaluation, let's briefly discuss what an exponential function is. An exponential function is a function of the form f(x) = a^x, where 'a' is a constant called the base, and 'x' is the exponent. The base 'a' must be a positive real number not equal to 1. In our case, the function is g(x) = 8^x, so the base is 8. The exponent 'x' can be any real number, which means we can plug in positive numbers, negative numbers, fractions, and even irrational numbers. Understanding this foundation is key to appreciating how exponential functions behave and how to manipulate them. The concept of the exponent is central – it dictates how many times the base is multiplied by itself (or, as we'll see with negative exponents, how many times we divide by the base). This simple yet powerful structure allows exponential functions to model rapid growth or decay, making them indispensable tools in various scientific and economic models. With this basic understanding, we're ready to explore the specific evaluation required in our problem.

Evaluating g(x) = 8^x at x = -2

Now, let's get to the heart of the matter: evaluating g(x) = 8^x when x = -2. This means we need to find the value of 8 raised to the power of -2. A negative exponent indicates a reciprocal. Specifically, a^(-n) is equal to 1/(a^n). Applying this rule to our problem, we have:

g(-2) = 8^(-2) = 1/(8^2)

This transformation is crucial because it converts the problem into a more manageable form. Instead of dealing with a negative exponent, we now have a positive exponent in the denominator. This is a common technique used in mathematics to simplify expressions and make calculations easier. The rule of negative exponents is not just a mathematical trick; it reflects a fundamental property of how exponents work and how they relate to division. By understanding this relationship, we can confidently manipulate expressions involving negative exponents and arrive at the correct solutions. This step-by-step approach ensures clarity and reduces the chance of errors. With this transformation, we are now ready to calculate the value of 8 squared and proceed with the division.

Next, we need to calculate 8 squared (8^2), which means 8 multiplied by itself:

8^2 = 8 * 8 = 64

So, our expression becomes:

g(-2) = 1/64

This is a straightforward calculation, but it's essential to be precise. Squaring a number is a fundamental operation, and in this context, it directly leads us to the denominator of our final answer. The ability to quickly and accurately perform these basic calculations is essential for success in mathematics. The result, 64, represents the square of the base, and it now forms the denominator of our fraction. We're almost there – we just need to perform the division to get the decimal representation of our answer. This final step will give us a clear, numerical value for g(-2), allowing us to easily interpret and use the result in further calculations or applications.

Converting to Decimal and Rounding

To get the final answer, we need to convert the fraction 1/64 into a decimal. To do this, we divide 1 by 64:

1 ÷ 64 = 0.015625

The problem asks us to round to four decimal places. Looking at the fifth decimal place (which is 2), we see that it is less than 5, so we round down and keep the fourth decimal place as it is.

g(-2) ≈ 0.0156

This rounding step is crucial for providing an answer in the requested format. It also highlights the practical aspect of mathematics, where results are often rounded for simplicity or to match the precision of measurements. The process of division may seem simple, but it's the bridge that connects the fractional representation to the decimal representation, allowing us to express the result in a more familiar and easily interpretable form. The decision to round down in this case is based on standard mathematical conventions, ensuring consistency and accuracy in our final answer. This rounded value is our final answer, representing the value of the function g(x) at x = -2, rounded to four decimal places.

Final Answer

Therefore, g(-2) ≈ 0.0156. This means that when we input -2 into the function g(x) = 8^x, the output is approximately 0.0156. This final answer encapsulates the entire process, from understanding the exponential function to applying the rules of exponents and performing the necessary calculations. It provides a concrete value for the function at the specified point, which can be used for further analysis or applications. The journey we've taken to arrive at this answer highlights the interconnectedness of mathematical concepts and the importance of a step-by-step approach. By carefully applying the rules and techniques we've discussed, we can confidently evaluate exponential functions and interpret their results. This understanding forms a solid foundation for tackling more complex mathematical problems and appreciating the power and versatility of exponential functions.

Conclusion

In summary, we have successfully evaluated the function g(x) = 8^x at x = -2 and found that g(-2) ≈ 0.0156. This exercise demonstrates the importance of understanding exponential functions and how to manipulate exponents, especially negative exponents. By following a clear, step-by-step approach, we can confidently solve these types of problems and gain a deeper appreciation for the elegance of mathematics. Remember the key concepts: the definition of exponential functions, the rule for negative exponents, and the process of converting fractions to decimals. These concepts are not only essential for this specific problem but also form the foundation for more advanced topics in mathematics. We encourage you to practice similar problems to solidify your understanding and build your mathematical skills. With consistent effort and a clear understanding of the underlying principles, you can master the evaluation of exponential functions and confidently apply them in various contexts.

For further learning and exploration of exponential functions, you can visit resources like Khan Academy's Exponential Functions section.