Finding F(-142) For F(x) = -0.2(x + 223) A Step-by-Step Guide
In the fascinating world of mathematics, functions serve as powerful tools that map inputs to outputs, unveiling the hidden relationships between variables. Understanding how to evaluate functions for specific inputs is a fundamental skill, allowing us to unlock the secrets they hold. This article delves into the process of finding the value of a function for a given input, using a concrete example to illustrate the steps involved.
The Challenge: Evaluating f(-142) for f(x) = -0.2(x + 223)
Our mission is to determine the value of the function f(x) = -0.2(x + 223) when the input x is -142. This means we need to substitute -142 for x in the function's expression and simplify the resulting expression.
The given function is f(x) = -0.2(x + 223). We are asked to find the value of f(-142). This requires us to substitute x = -142 into the function. So, we replace x with -142 in the equation: f(-142) = -0.2(-142 + 223).
Step-by-Step Solution: Unveiling the Value
Let's embark on a step-by-step journey to solve this problem:
- Substitution: The first step involves replacing the variable 'x' in the function's expression with the given input value, which is -142. This transforms the expression f(x) = -0.2(x + 223) into f(-142) = -0.2(-142 + 223).
- Simplification within Parentheses: Next, we focus on simplifying the expression inside the parentheses. Adding -142 and 223 results in 81. So, our expression becomes f(-142) = -0.2(81).
- Multiplication: Now, we perform the multiplication operation. Multiplying -0.2 by 81 yields -16.2. Therefore, f(-142) = -16.2.
Detailed Breakdown of the Calculation
To ensure clarity, let's break down the calculation step-by-step:
- Original Function: f(x) = -0.2(x + 223)
- Substitute x = -142: f(-142) = -0.2(-142 + 223)
- Simplify inside the parentheses: -142 + 223 = 81, so f(-142) = -0.2(81)
- Multiply -0.2 by 81: -0. 2 * 81 = -16.2
- Final Result: f(-142) = -16.2
The Answer Revealed: f(-142) = -16.2
Through our meticulous calculations, we have arrived at the solution: f(-142) = -16.2. This means that when the input to the function f(x) is -142, the corresponding output is -16.2. This corresponds to option D.
Therefore, the correct answer is:
D. -16.2
Grasping the Significance: Function Evaluation
Evaluating functions is a cornerstone of mathematical analysis. It allows us to determine the output of a function for any given input, providing valuable insights into the function's behavior and the relationship between variables. By mastering the art of function evaluation, we unlock the power to predict, model, and understand a wide range of phenomena.
Mastering Function Evaluation: A Key to Mathematical Proficiency
Function evaluation is not just a mathematical exercise; it's a fundamental skill that empowers us to solve real-world problems. Whether we're modeling population growth, analyzing financial data, or designing engineering systems, the ability to evaluate functions accurately is essential for success. By diligently practicing this skill, we pave the way for deeper mathematical understanding and enhanced problem-solving capabilities.
The Importance of Precision: Avoiding Errors in Function Evaluation
In the realm of function evaluation, precision is paramount. A single misstep in calculation can lead to an incorrect result, potentially undermining the entire analysis. Therefore, it's crucial to pay meticulous attention to detail, double-checking each step and ensuring that every operation is performed accurately. By cultivating a habit of precision, we safeguard against errors and ensure the reliability of our findings.
In multiple-choice questions, selecting the correct answer often hinges on accurate calculation and careful attention to detail. In our example, we had four options to choose from, and only one of them, -16.2, matched the value we calculated for f(-142). This highlights the importance of not only understanding the mathematical concepts but also executing the calculations flawlessly.
Strategies for Success: Tackling Function Evaluation Questions
When faced with function evaluation questions, a systematic approach can be your greatest ally. Start by carefully reading the question and identifying the function and the input value. Then, substitute the input value into the function's expression and simplify the resulting expression step-by-step. Finally, compare your answer with the given options and select the one that matches. By following these strategies, you can confidently navigate function evaluation questions and arrive at the correct solution.
Beyond the Numbers: The Power of Functions
While function evaluation is a crucial skill, it's important to remember that functions are more than just mathematical expressions. They are powerful tools that allow us to model and understand the world around us. From predicting weather patterns to designing bridges, functions play a vital role in countless applications. By appreciating the versatility of functions, we gain a deeper understanding of their significance and their ability to shape our world.
Conclusion: The Art of Function Evaluation
In this article, we have explored the process of evaluating functions, using the example of f(x) = -0.2(x + 223) to illustrate the steps involved. We have seen how substituting the input value, simplifying the expression, and performing the necessary calculations lead us to the desired output. By mastering the art of function evaluation, we unlock a powerful tool for mathematical analysis and problem-solving, paving the way for deeper insights and a greater appreciation of the world around us.
