Evaluating Functions Finding F(-142) For F(x) = -0.2(x + 223)

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In the realm of mathematics, functions serve as fundamental building blocks, mapping inputs to corresponding outputs. Function evaluation, a cornerstone of mathematical analysis, involves substituting a specific value for the variable in a function's expression to determine the output. This process allows us to understand the behavior of functions and their applications in various contexts. In this article, we will embark on a journey to evaluate a given function, f(x) = -0.2(x + 223), at a specific point, x = -142. This exploration will not only solidify our understanding of function evaluation but also showcase its practical relevance in solving mathematical problems.

The function we are tasked with evaluating is f(x) = -0.2(x + 223). This function takes an input x, adds 223 to it, and then multiplies the result by -0.2. To find the value of f(-142), we will substitute -142 for x in the function's expression. This substitution is the heart of function evaluation, allowing us to transform a general function into a specific numerical value. By carefully performing this substitution, we can unlock the output of the function at the given input, providing valuable insights into the function's behavior at that particular point.

Now, let's delve into the step-by-step process of evaluating f(-142). We begin by substituting x = -142 into the function's expression: f(-142) = -0.2(-142 + 223). The next step involves simplifying the expression inside the parentheses: -142 + 223 = 81. This simplification is crucial as it reduces the complexity of the expression, making it easier to perform the subsequent calculations. With the expression inside the parentheses simplified, we can now proceed to the final step: multiplying -0.2 by 81. This multiplication will yield the value of f(-142), the output of the function at the input x = -142. Through this meticulous process, we transform the function evaluation problem into a straightforward arithmetic calculation, highlighting the power of function evaluation in extracting specific information from a function's expression.

Performing the multiplication, we get: f(-142) = -0.2 * 81 = -16.2. Therefore, the value of the function f(x) = -0.2(x + 223) at x = -142 is -16.2. This result provides us with a specific point on the function's graph, which can be visualized as the coordinate (-142, -16.2). This coordinate represents the output of the function when the input is -142, showcasing the direct relationship between input and output that functions define. Understanding this relationship is paramount in various mathematical and real-world applications, where functions are used to model and predict phenomena.

In conclusion, we have successfully evaluated the function f(x) = -0.2(x + 223) at x = -142, obtaining the value -16.2. This process involved substituting the given value for the variable, simplifying the expression, and performing the necessary calculations. Function evaluation is a fundamental skill in mathematics, enabling us to understand the behavior of functions and their applications in diverse fields. By mastering this skill, we equip ourselves with a powerful tool for problem-solving and analysis.

To solidify our understanding, let's break down the solution into a detailed step-by-step process. This meticulous approach will not only reinforce the concept of function evaluation but also provide a clear roadmap for tackling similar problems in the future. Each step will be carefully explained, ensuring that the underlying logic is fully grasped. This step-by-step solution serves as a valuable resource for anyone seeking to master function evaluation.

Step 1: Substitute x = -142 into the function. The first step in evaluating f(-142) is to substitute -142 for x in the function's expression, f(x) = -0.2(x + 223). This substitution transforms the general function into a specific expression that can be evaluated numerically. The result of this substitution is: f(-142) = -0.2(-142 + 223). This step is the cornerstone of function evaluation, as it sets the stage for the subsequent calculations.

Step 2: Simplify the expression inside the parentheses. Next, we simplify the expression inside the parentheses: -142 + 223. This simplification involves basic arithmetic, adding two numbers with different signs. The result of this addition is: -142 + 223 = 81. This step reduces the complexity of the expression, making it easier to perform the final calculation. By simplifying the expression, we streamline the evaluation process and minimize the chances of errors.

Step 3: Multiply -0.2 by 81. The final step is to multiply -0.2 by 81. This multiplication will yield the value of f(-142). The result of this multiplication is: -0.2 * 81 = -16.2. This step completes the evaluation process, providing us with the output of the function at the input x = -142. By performing this calculation, we arrive at the solution to the problem.

Therefore, f(-142) = -16.2. This step-by-step solution provides a clear and concise pathway to evaluating the function at the given point. By understanding each step and the reasoning behind it, we can confidently tackle similar function evaluation problems in the future. This methodical approach ensures accuracy and fosters a deeper understanding of function evaluation.

While function evaluation is a fundamental concept, it is not immune to errors. Understanding common mistakes can significantly improve accuracy and prevent frustration. Let's explore some pitfalls to avoid when evaluating functions. These insights will help you navigate the process with greater confidence and precision.

One common mistake is incorrect substitution. This can occur when the value being substituted is not placed correctly in the function's expression. For instance, in our example, if -142 were not substituted correctly for x in the expression f(x) = -0.2(x + 223), the entire evaluation would be flawed. To avoid this, carefully double-check the substitution before proceeding with the calculations. Ensure that the value being substituted replaces the variable accurately throughout the expression. This meticulous approach will minimize the risk of errors stemming from incorrect substitution.

Another frequent error is incorrect order of operations. Mathematical expressions must be simplified following the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Failing to adhere to this order can lead to incorrect results. In our example, the expression inside the parentheses, -142 + 223, must be simplified before multiplying by -0.2. Neglecting this order would yield an incorrect answer. To avoid this mistake, always prioritize operations within parentheses, followed by exponents, multiplication and division (from left to right), and finally, addition and subtraction (from left to right). This disciplined approach ensures that expressions are simplified correctly, leading to accurate evaluations.

Arithmetic errors are also a common source of mistakes in function evaluation. These errors can occur during any of the arithmetic operations involved, such as addition, subtraction, multiplication, or division. In our example, an arithmetic error in simplifying -142 + 223 or in multiplying -0.2 by 81 would lead to an incorrect final answer. To minimize these errors, double-check each calculation carefully. Consider using a calculator for complex calculations or for those prone to errors. By paying close attention to arithmetic details, you can significantly reduce the likelihood of making mistakes during function evaluation.

Finally, sign errors can be tricky pitfalls in function evaluation. These errors often occur when dealing with negative numbers. In our example, the negative sign in -0.2 and the negative sign in -142 can potentially lead to sign errors if not handled carefully. To avoid these errors, pay close attention to the signs of numbers throughout the evaluation process. Remember the rules for multiplying and dividing numbers with different signs (a negative times a positive is a negative, and vice versa). By being mindful of signs, you can avoid these common errors and ensure accurate function evaluation.

By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy in function evaluation. Careful substitution, adherence to the order of operations, meticulous arithmetic, and attention to signs are key to mastering this fundamental mathematical skill.

Practice is the key to mastering any mathematical concept, and function evaluation is no exception. To solidify your understanding and hone your skills, let's explore a few practice problems. These problems will provide you with the opportunity to apply the concepts we've discussed and identify any areas that may require further attention. By working through these exercises, you'll gain confidence in your ability to evaluate functions accurately and efficiently.

Problem 1: Evaluate g(x) = 3x^2 - 2x + 1 at x = -2.

This problem introduces a quadratic function, which includes a squared term. The process of evaluation remains the same: substitute the given value for x and simplify the expression using the order of operations. This problem will challenge your ability to handle exponents and multiple arithmetic operations.

Solution:

  1. Substitute: g(-2) = 3(-2)^2 - 2(-2) + 1
  2. Exponents: g(-2) = 3(4) - 2(-2) + 1
  3. Multiplication: g(-2) = 12 + 4 + 1
  4. Addition: g(-2) = 17

Therefore, g(-2) = 17.

Problem 2: Evaluate h(x) = (x + 5) / (x - 3) at x = 1.

This problem involves a rational function, which includes a fraction. The key here is to simplify the numerator and denominator separately before performing the division. This problem will test your ability to work with fractions and handle potential division-by-zero scenarios (which are not present in this particular case, but something to keep in mind).

Solution:

  1. Substitute: h(1) = (1 + 5) / (1 - 3)
  2. Simplify Numerator: h(1) = 6 / (1 - 3)
  3. Simplify Denominator: h(1) = 6 / (-2)
  4. Division: h(1) = -3

Therefore, h(1) = -3.

Problem 3: Evaluate k(x) = √(x + 9) at x = 7.

This problem introduces a square root function. Remember that the square root of a number is a value that, when multiplied by itself, equals the original number. This problem will test your understanding of square roots and your ability to handle them in function evaluation.

Solution:

  1. Substitute: k(7) = √(7 + 9)
  2. Simplify: k(7) = √16
  3. Square Root: k(7) = 4

Therefore, k(7) = 4.

By working through these practice problems, you'll gain valuable experience in evaluating various types of functions. Remember to carefully substitute, follow the order of operations, and double-check your calculations. The more you practice, the more confident and proficient you'll become in function evaluation.

In conclusion, function evaluation is a fundamental concept in mathematics that allows us to determine the output of a function for a given input. We have explored the step-by-step process of evaluating a function, from substituting the input value to simplifying the expression and arriving at the final result. We have also discussed common mistakes to avoid, such as incorrect substitution, errors in the order of operations, arithmetic errors, and sign errors. By being aware of these pitfalls, you can significantly improve your accuracy in function evaluation.

Furthermore, we have worked through several practice problems involving different types of functions, including quadratic, rational, and square root functions. These exercises have provided you with the opportunity to apply the concepts we've discussed and solidify your understanding of function evaluation. Remember that practice is key to mastery, so continue to work through additional problems to hone your skills.

The ability to evaluate functions accurately and efficiently is essential for success in various mathematical and scientific fields. Functions are used to model real-world phenomena, make predictions, and solve complex problems. By mastering function evaluation, you are equipping yourself with a powerful tool that will serve you well in your academic and professional endeavors.

As you continue your mathematical journey, remember that function evaluation is not just a skill to be memorized, but a concept to be understood. Strive to grasp the underlying logic and reasoning behind each step in the process. This deep understanding will enable you to tackle even the most challenging function evaluation problems with confidence and precision. With dedication and practice, you can master function evaluation and unlock its full potential.