Evaluating The Function G(x) = (1/4)^x At X = 2.14

In this article, we will delve into the evaluation of an exponential function. Exponential functions are a fundamental concept in mathematics, with widespread applications in various fields such as finance, physics, and computer science. They model phenomena that exhibit exponential growth or decay. Our focus will be on the specific function g(x) = (1/4)^x, and we will meticulously calculate its value at x = 2.14. This exercise will provide a practical understanding of how to work with exponential functions and how to arrive at accurate numerical solutions. Understanding exponential functions and their evaluation is crucial for anyone seeking to grasp the mathematical underpinnings of many real-world phenomena. The exponential function is defined as f(x) = a^x, where a is a positive constant (the base) and x is the variable exponent. When the base a is between 0 and 1, the function represents exponential decay; when a is greater than 1, it represents exponential growth. In our case, the base is 1/4, which is between 0 and 1, so we are dealing with an exponential decay function. This means that as x increases, the value of g(x) decreases. The importance of understanding exponential functions extends beyond theoretical mathematics. They are used to model compound interest, population growth, radioactive decay, and many other real-world processes. Accurately evaluating exponential functions is essential for making predictions and decisions in these fields. For instance, in finance, exponential functions are used to calculate the future value of investments. In physics, they are used to describe the decay of radioactive isotopes. In computer science, they are used in the analysis of algorithms and data structures. Our evaluation of g(2.14) will not only provide a numerical answer but also reinforce the understanding of the behavior of exponential functions and their role in mathematical modeling. We will approach this calculation with a focus on accuracy and clarity, demonstrating the steps involved in evaluating an exponential function with a decimal exponent. This hands-on approach is vital for building confidence and competence in handling such calculations. Exponential functions are also closely related to logarithmic functions, which are their inverses. Understanding one helps in understanding the other. The logarithm of a number x to the base a is the exponent to which a must be raised to produce x. This relationship is expressed as log_a(x) = y if and only if a^y = x. These functions are used together in many applications, particularly in solving equations where the variable is in the exponent. Therefore, mastering exponential functions is a stepping stone to mastering logarithms and their applications as well. In the subsequent sections, we will proceed with the actual calculation, breaking down the steps to ensure clarity and accuracy. This practical demonstration will solidify the concepts discussed here and provide a useful reference for similar evaluations in the future.

Problem Statement

The problem at hand is to find the value of the function g(x) = (1/4)^x when x = 2.14. This requires us to substitute 2.14 for x in the function's expression and then compute the resulting value. The expression (1/4)^2.14 represents the number 1/4 raised to the power of 2.14. Since 2.14 is not an integer, we will need to use a calculator or mathematical software to obtain an accurate result. Before we dive into the calculation, let's briefly discuss why this type of problem is significant. Evaluating functions at specific points is a core skill in mathematics. It allows us to understand the behavior of the function and its output for different inputs. In the case of exponential functions, this skill is particularly important because these functions often model real-world phenomena, as mentioned earlier. By evaluating g(2.14), we are essentially finding the value of a quantity that decays exponentially, such as the amount of a radioactive substance remaining after a certain time or the value of an investment that depreciates at a constant rate. The exponent 2.14 itself is a decimal, which adds a layer of complexity to the calculation. Raising a number to a non-integer power involves the use of fractional exponents and can be understood in terms of roots and powers. For example, a^(m/n) can be expressed as the n-th root of a raised to the power of m, or (a^(1/n))m. In our case, we can think of (1/4)^2.14 as (1/4)^(214/100), which could theoretically be calculated by finding the 100th root of (1/4)^214. However, this is not a practical approach for manual calculation, and it is much more efficient to use a calculator or software that can handle decimal exponents directly. The problem also highlights the importance of rounding in mathematical calculations. Since we are asked to round the result to the nearest thousandth, we need to consider the digit in the ten-thousandths place to ensure that our rounding is accurate. This step is crucial for providing a meaningful and practical answer. In summary, the problem of finding g(2.14) for the function g(x) = (1/4)^x is a straightforward yet important exercise in function evaluation. It tests our understanding of exponential functions, decimal exponents, and the use of calculators or software for accurate computation. By solving this problem, we reinforce our ability to apply mathematical concepts to practical situations and to interpret the results in a meaningful way. The following sections will detail the step-by-step calculation and provide the final answer rounded to the nearest thousandth.

Step-by-Step Solution

To find g(2.14) for the function g(x) = (1/4)^x, we need to substitute x = 2.14 into the expression. This gives us g(2.14) = (1/4)^2.14. As discussed earlier, we will use a calculator to evaluate this expression. Most scientific calculators and software platforms have the capability to handle exponents, including decimal exponents. The process typically involves entering the base (1/4 or 0.25), then using the exponentiation key (often denoted as x^y or ^), and finally entering the exponent (2.14). When you input this into a calculator, you should get a result close to 0.0569655. However, we are asked to round the answer to the nearest thousandth. The thousandth place is the third digit after the decimal point. In our result, this is the digit 6. To determine whether to round up or down, we look at the digit in the ten-thousandths place, which is 9. Since 9 is greater than or equal to 5, we round up the digit in the thousandth place. This means that 6 becomes 7. Therefore, the value of g(2.14) rounded to the nearest thousandth is 0.057. It is important to understand the concept of rounding and its implications. Rounding is a way of simplifying a number while preserving its approximate value. It is often necessary when dealing with decimal numbers that have many digits, especially in practical applications where excessive precision may not be meaningful or required. Rounding to the nearest thousandth provides a reasonable level of accuracy for most purposes. It is also important to be aware of the potential for rounding errors. Rounding at intermediate steps in a calculation can lead to a slightly different final result than if the rounding is done only at the end. Therefore, it is generally recommended to perform calculations with as much precision as possible and round only the final answer. In this specific problem, we have rounded the result of (1/4)^2.14 to 0.057. This means that the true value of g(2.14) is somewhere between 0.0565 and 0.0575. This level of accuracy is sufficient for most practical purposes. In summary, the step-by-step solution involves substituting the value of x into the function, using a calculator to evaluate the expression, and rounding the result to the specified level of precision. The key steps are: 1. Substitute x = 2.14 into g(x) = (1/4)^x to get g(2.14) = (1/4)^2.14. 2. Use a calculator to evaluate (1/4)^2.14, which yields approximately 0.0569655. 3. Round the result to the nearest thousandth, which gives us 0.057. This process demonstrates the practical application of exponential functions and the importance of accurate calculation and rounding. The final answer provides a clear and concise solution to the problem.

Calculation and Result

As we established in the previous section, the core of our problem lies in calculating (1/4)^2.14. To achieve this, we utilize a scientific calculator, which is equipped to handle exponential operations with decimal exponents. Inputting the expression (1/4)^2.14 into the calculator yields a result of approximately 0.0569655. This raw result contains more digits than are necessary for our purpose, as we are required to round to the nearest thousandth. The concept of significant figures comes into play here. When we round a number, we are essentially retaining only the digits that are considered significant, meaning they contribute to the precision of the number. In this case, rounding to the nearest thousandth means we want to keep three digits after the decimal point. As we examine the result 0.0569655, we focus on the thousandth's place, which is occupied by the digit 6. The digit immediately to the right, in the ten-thousandth's place, is 9. The rule for rounding states that if the digit to the right of the rounding place is 5 or greater, we round up; otherwise, we round down. Since 9 is greater than 5, we round the 6 in the thousandth's place up to 7. Consequently, the value of (1/4)^2.14 rounded to the nearest thousandth is 0.057. This seemingly simple rounding step is crucial in ensuring the accuracy and clarity of our final answer. It is a common practice in scientific and engineering calculations to round results to an appropriate number of significant figures, as it reflects the precision of the measurements or calculations involved. In our case, rounding to the nearest thousandth provides a balance between precision and simplicity. The final result, 0.057, represents the value of the function g(x) = (1/4)^x at x = 2.14, rounded to the nearest thousandth. This value tells us that when we raise 1/4 to the power of 2.14, we obtain a number that is very close to 0.057. It is worth noting that exponential functions with a base between 0 and 1, like our g(x), exhibit exponential decay. This means that as the exponent x increases, the value of the function decreases. Our result confirms this behavior, as (1/4)^2.14 is significantly smaller than (1/4)^0 = 1 or (1/4)^1 = 0.25. In summary, the calculation and rounding process has led us to the final result of 0.057. This value is the answer to our problem, representing the value of the exponential function g(x) = (1/4)^x at x = 2.14, rounded to the nearest thousandth. This result is not only a numerical answer but also a demonstration of the practical application of exponential functions and the importance of accurate calculations and rounding in mathematics.

Final Answer

After performing the calculation and rounding to the nearest thousandth, we have determined that g(2.14) = 0.057. This is the final answer to the problem. To reiterate, this means that when the function g(x) = (1/4)^x is evaluated at x = 2.14, the resulting value is approximately 0.057. This result is a decimal number less than 1, which is consistent with the behavior of an exponential decay function, as the base (1/4) is between 0 and 1. The value 0.057 represents a specific point on the graph of the function g(x). If we were to plot the graph, the point (2.14, 0.057) would lie on the curve. This point gives us a visual representation of the function's behavior at x = 2.14. The final answer is not just a number; it is a piece of information that contributes to our understanding of the function g(x) and its properties. It demonstrates the concept of exponential decay and provides a concrete example of the function's output for a particular input. In the context of real-world applications, this result could represent a quantity that is decreasing exponentially, such as the amount of a radioactive substance remaining after a certain time or the value of an asset that is depreciating. The specific meaning would depend on the context in which the function g(x) is being used. The process of arriving at the final answer has involved several steps, including substituting the value of x into the function, using a calculator to evaluate the expression, and rounding the result to the nearest thousandth. Each of these steps is important for ensuring the accuracy and clarity of the final answer. The rounding step, in particular, highlights the importance of providing answers with an appropriate level of precision. Rounding to the nearest thousandth is a common practice in mathematics and science, as it provides a balance between accuracy and simplicity. In conclusion, the final answer to the problem is g(2.14) = 0.057. This result is a clear and concise solution that answers the question posed in the problem statement. It demonstrates our understanding of exponential functions, function evaluation, and rounding techniques. The final answer serves as a valuable piece of information about the function g(x) and its behavior at a specific point.