Finding F(-17.1) For F(x) = Cube Root Of -2x + 3

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Understanding the Function and the Problem

In this article, we will delve into the process of evaluating a cube root function for a specific input value. The given function is f(x) = \sqrt[3]{-2x + 3}, and our goal is to find the value of f(-17.1). This involves substituting -17.1 for x in the function and simplifying the expression to arrive at the final answer in decimal notation. This task requires a solid understanding of function evaluation and order of operations, as well as familiarity with cube roots and decimal arithmetic. By meticulously working through each step, we can accurately determine the value of f(-17.1) and gain a deeper appreciation for the behavior of cube root functions. This exercise not only provides a solution to a specific problem but also reinforces essential mathematical skills applicable in various contexts. Functions play a vital role in mathematics and its applications, enabling us to model real-world phenomena and solve complex problems. Mastering the evaluation of functions is therefore a fundamental skill for anyone pursuing further studies in mathematics, science, or engineering. Furthermore, the application of order of operations and decimal arithmetic in this problem highlights the importance of precision and attention to detail in mathematical calculations. By carefully performing each step and verifying the result, we can ensure the accuracy of our solution and gain confidence in our mathematical abilities. Ultimately, this exercise serves as a valuable learning experience, enhancing our understanding of functions, cube roots, and decimal arithmetic, and preparing us for more advanced mathematical challenges.

Step-by-Step Solution

To find f(-17.1), we substitute -17.1 for x in the function f(x) = \sqrt[3]{-2x + 3}:

  1. Substitution:
    • f(-17.1) = \sqrt[3]{-2(-17.1) + 3}

The first step in evaluating the function f(x) = \sqrt[3]{-2x + 3} at x = -17.1 is to substitute the value -17.1 for x in the function. This means replacing every instance of the variable x in the function's expression with the value -17.1. This substitution is a fundamental operation in function evaluation, allowing us to determine the output of the function for a given input. The accuracy of the subsequent steps depends heavily on the correct substitution of the input value. In this case, substituting -17.1 for x gives us the expression \sqrt[3]{-2(-17.1) + 3}. This expression now represents the value of the function f(x) at x = -17.1, and our goal is to simplify it to obtain a numerical answer. The next step involves performing the arithmetic operations within the cube root, following the order of operations (PEMDAS/BODMAS). This will involve multiplication and addition, which must be carried out in the correct sequence to arrive at the correct result. By carefully substituting the value and preparing for the subsequent arithmetic operations, we set the stage for a successful evaluation of the function.

  1. Multiplication:
    • -2 * -17.1 = 34.2
    • f(-17.1) = \sqrt[3]{34.2 + 3}

After substituting -17.1 for x in the function, the next step is to perform the multiplication within the cube root. We have -2 multiplied by -17.1, which results in a positive value of 34.2. It is crucial to pay attention to the signs during multiplication; a negative number multiplied by a negative number yields a positive number. This step is essential for simplifying the expression inside the cube root and moving closer to the final answer. The multiplication operation is a fundamental arithmetic operation, and its accurate execution is critical for the correctness of the entire calculation. By performing the multiplication correctly, we reduce the complexity of the expression and prepare for the next operation, which is addition. The result of the multiplication, 34.2, will be added to 3 in the subsequent step. This process of simplifying the expression step by step ensures that we follow the order of operations and arrive at the correct value for f(-17.1). Furthermore, this step highlights the importance of precision in decimal arithmetic. The accurate multiplication of -2 and -17.1 is crucial for obtaining the correct decimal value, which will then affect the final result. By carefully performing the multiplication and verifying the result, we maintain the integrity of the calculation and ensure the accuracy of our final answer.

  1. Addition:
    • 34.2 + 3 = 37.2
    • f(-17.1) = \sqrt[3]{37.2}

Following the multiplication, the next step in simplifying the expression is to perform the addition within the cube root. We have 34.2 plus 3, which equals 37.2. This addition operation combines the two terms inside the cube root into a single value, making the expression more manageable. The accuracy of this addition is crucial for obtaining the correct final result. Similar to the multiplication step, attention to decimal arithmetic is essential to ensure the precision of the calculation. Adding 3 to 34.2 is a straightforward operation, but care must be taken to align the decimal points correctly and avoid errors. By accurately performing the addition, we arrive at the simplified expression \sqrt[3]{37.2}. This expression represents the cube root of 37.2, which is the final operation we need to perform to evaluate the function f(-17.1). The cube root of 37.2 is the number that, when multiplied by itself three times, equals 37.2. Finding this value will give us the numerical answer for f(-17.1). The addition step is an important intermediate step in the overall calculation, reducing the complexity of the expression and bringing us closer to the final solution. By carefully performing the addition and verifying the result, we ensure the integrity of the calculation and prepare for the final cube root operation.

  1. Cube Root:
    • \sqrt[3]{37.2} ≈ 3.336
    • f(-17.1) ≈ 3.336

After simplifying the expression inside the cube root, the final step is to calculate the cube root of 37.2. The cube root of a number is the value that, when multiplied by itself three times, equals the original number. In this case, we need to find the number that, when cubed, gives us 37.2. This calculation can be done using a calculator or a numerical method. The cube root of 37.2 is approximately 3.336. This value is an approximation because the cube root of 37.2 is an irrational number, meaning its decimal representation goes on infinitely without repeating. Therefore, we round the result to a suitable number of decimal places, in this case, three decimal places. This approximation is accurate enough for most practical purposes. The cube root operation is the inverse of cubing a number, and it is a fundamental operation in mathematics. Understanding cube roots is essential for solving various types of equations and problems. In the context of this problem, finding the cube root of 37.2 gives us the final value of the function f(-17.1). By accurately calculating the cube root and rounding the result appropriately, we arrive at the solution to the problem. The final answer, f(-17.1) ≈ 3.336, represents the value of the function f(x) = \sqrt[3]{-2x + 3} when x is equal to -17.1. This result provides a numerical answer to the problem and demonstrates the process of evaluating a cube root function for a specific input value.

Final Answer

Therefore, f(-17.1) ≈ 3.336.

In conclusion, by following the order of operations and performing the necessary arithmetic calculations, we have successfully determined the value of f(-17.1) for the function f(x) = \sqrt[3]{-2x + 3}. The final answer, approximately 3.336, provides a numerical solution to the problem and demonstrates the process of evaluating a cube root function. This exercise reinforces the importance of function evaluation, order of operations, and decimal arithmetic in mathematics. By carefully working through each step and verifying the result, we can ensure the accuracy of our solution and gain confidence in our mathematical abilities.