Evaluating F(4) For The Function F(x) = 3.1(3)^(-x+1) + 1.6

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Introduction

In mathematics, functions are fundamental tools for modeling relationships between variables. Evaluating a function at a specific point involves substituting the given value for the variable and simplifying the expression. In this article, we will focus on evaluating the function f(x) = 3.1(3)^{-x+1} + 1.6 at x = 4. We will walk through the step-by-step process, ensuring accuracy and clarity in our calculations. This exercise not only reinforces the concept of function evaluation but also highlights the importance of order of operations and decimal precision in mathematical computations. Let's dive into the world of functions and explore how we can find the value of f(4) for the given function.

Understanding the Function

The function we are dealing with is f(x) = 3.1(3)^{-x+1} + 1.6. This function combines several mathematical operations, including exponentiation, multiplication, addition, and constants. To accurately evaluate this function, it is essential to understand the order of operations, often remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). The function consists of a constant term (1.6) and a term involving an exponential expression (3.1(3)^{-x+1}). The exponential part has a base of 3 and an exponent of -x+1, which means the value of the exponent changes based on the input x. The coefficient 3.1 scales this exponential term. To evaluate f(x) at x = 4, we will substitute 4 for x in the expression and follow the order of operations to simplify. This careful approach will lead us to the correct result, ensuring we understand how each part of the function contributes to its overall value at a given point.

Step-by-Step Evaluation of f(4)

To evaluate the function f(x) = 3.1(3)^{-x+1} + 1.6 at x = 4, we need to substitute 4 for x in the function and simplify the expression step-by-step.

  1. Substitution: Replace x with 4 in the function: f(4) = 3.1(3)^{-4+1} + 1.6

  2. Simplify the Exponent: Simplify the exponent -4 + 1: f(4) = 3.1(3)^{-3} + 1.6

  3. Evaluate the Exponential Term: Calculate 3 raised to the power of -3: 3^{-3} = 1 / 3^3 = 1 / 27

    So, the function becomes: f(4) = 3.1(1/27) + 1.6

  4. Perform Multiplication: Multiply 3.1 by 1/27: 3. 1 * (1/27) ≈ 0.1148

    The function now looks like this: f(4) ≈ 0.1148 + 1.6

  5. Perform Addition: Add 0.1148 to 1.6: f(4) ≈ 1.7148

Therefore, the value of the function f(x) at x = 4 is approximately 1.7148. This step-by-step breakdown ensures accuracy and clarity in the evaluation process, highlighting how each operation contributes to the final result. By meticulously following these steps, we have successfully evaluated f(4) and can confidently present our answer.

Rounding to Four Decimal Places

As specified in the problem statement, we need to round our answer to four decimal places. Our calculated value for f(4) is approximately 1.7148. To round this to four decimal places, we look at the fifth decimal place, which in this case is non-existent, implying it is 0. Since there is nothing beyond the fourth decimal place, we don't need to round up. Thus, the rounded value remains 1.7148. Rounding to a specific number of decimal places is crucial in many mathematical and scientific contexts to maintain a consistent level of precision and avoid misinterpretations. In this instance, our result of 1.7148 represents the function's value at x = 4 with the required accuracy, ensuring that our final answer adheres to the problem's constraints.

Final Answer

After evaluating the function f(x) = 3.1(3)^{-x+1} + 1.6 at x = 4 and rounding the result to four decimal places, we have arrived at our final answer: f(4) ≈ 1.7148. This result signifies the function's output when the input is 4. By systematically following the order of operations, simplifying the expression, and applying the necessary rounding, we have successfully determined the value of the function at the given point. This comprehensive process demonstrates a clear understanding of function evaluation and the importance of precision in mathematical calculations. Our final answer provides a precise numerical representation of the function's behavior at x = 4, fulfilling the requirements of the problem statement.

Conclusion

In conclusion, we have successfully evaluated the function f(x) = 3.1(3)^{-x+1} + 1.6 at x = 4. By following a step-by-step approach, which included substitution, simplification of the exponent, evaluation of the exponential term, multiplication, and addition, we found that f(4) is approximately equal to 1.7148. We also emphasized the importance of rounding the answer to four decimal places as per the instructions. This exercise underscores the fundamental principles of function evaluation and the significance of precision in mathematical calculations. The methodical approach we took ensures that the result is accurate and reliable. Evaluating functions is a crucial skill in mathematics and has numerous applications in various fields, including physics, engineering, and computer science. This detailed walkthrough serves as a valuable reference for anyone looking to enhance their understanding of function evaluation and mathematical problem-solving.