If A Function Is Defined As F(x) = X/2 + 8, What Is The Value Of F(x) When X = 10?

Introduction

In mathematics, functions are fundamental tools for modeling relationships between variables. Understanding how to evaluate functions for specific inputs is a crucial skill. This article will delve into the process of evaluating the function f(x) = x/2 + 8 when x = 10. We will break down the steps involved, explore the underlying concepts, and provide a comprehensive explanation to ensure clarity and understanding. Whether you're a student learning about functions for the first time or simply looking to refresh your knowledge, this guide will provide you with a solid foundation. This exploration will not only help in solving this particular problem but also in grasping the broader concept of function evaluation, which is a cornerstone of mathematical analysis.

Understanding the Function Notation

Before we dive into the calculation, let's first understand the function notation f(x) = x/2 + 8. This notation tells us that we have a function named "f" that takes an input value, represented by the variable "x", and performs a specific operation on it. In this case, the function f(x) takes the input x, divides it by 2, and then adds 8 to the result. The expression x/2 + 8 is the rule or formula that defines how the function transforms the input x. Understanding function notation is essential because it allows us to express complex mathematical relationships in a concise and clear manner. It is a symbolic language that mathematicians use to communicate ideas and perform calculations. The function f(x) can be visualized as a machine that takes an input x, processes it according to the rule x/2 + 8, and produces an output. This output is the value of the function at that particular input x. Function notation is not just a shorthand; it's a powerful way of thinking about mathematical relationships and how they behave.

Evaluating f(x) when x = 10

To find the value of f(x) when x = 10, we need to substitute 10 for x in the function's formula. This means replacing every instance of x in the expression x/2 + 8 with the number 10. So, we have f(10) = 10/2 + 8. Now, we follow the order of operations (PEMDAS/BODMAS), which dictates that we perform division before addition. First, we divide 10 by 2, which gives us 5. Therefore, the expression becomes f(10) = 5 + 8. Next, we add 5 and 8, which equals 13. Thus, f(10) = 13. This means that when the input x is 10, the output of the function f(x) is 13. Evaluating functions is a fundamental skill in mathematics, and this example demonstrates a straightforward application of this concept. By substituting the given value into the function's formula and following the order of operations, we can determine the function's output for that specific input. This process is used extensively in various mathematical contexts, including calculus, algebra, and real-world applications.

Step-by-Step Solution

Let's break down the solution into a step-by-step process for clarity:

1. Write down the function: f(x) = x/2 + 8
2. Substitute x with 10: f(10) = 10/2 + 8
3. Perform the division: f(10) = 5 + 8
4. Perform the addition: f(10) = 13

Therefore, when x = 10, f(x) = 13. This step-by-step approach highlights the importance of following the correct order of operations to arrive at the correct answer. Each step builds upon the previous one, ensuring a logical and systematic solution. This method is not only applicable to this specific problem but also to a wide range of mathematical problems involving function evaluation. By breaking down the problem into smaller, manageable steps, we can avoid errors and gain a deeper understanding of the underlying concepts. This approach fosters a methodical way of problem-solving, which is a valuable skill in mathematics and beyond.

Choosing the Correct Answer

Looking at the given options:

A. 4 B. 9 C. 13 D. 36

The correct answer is C. 13, as we calculated f(10) = 13. This step involves comparing the result we obtained through our calculations with the provided options. It's a crucial step in any problem-solving process to ensure that we select the correct answer. In this case, we found that the value of f(x) when x = 10 is 13, which matches option C. This reinforces the accuracy of our calculations and our understanding of the problem. This final check is an important practice to minimize errors and build confidence in our solutions. It also highlights the importance of paying attention to detail and carefully reviewing the answer choices to select the one that aligns with our calculated result.

Why Other Options Are Incorrect

It's helpful to understand why the other options are incorrect. This can solidify your understanding of the function and the evaluation process:

• A. 4: This answer might be obtained if one mistakenly subtracts 8 from the result of 10/2 instead of adding.
• B. 9: This answer could be the result of adding 8 to 1 (perhaps thinking of 10/2 as 1 instead of 5), demonstrating a misunderstanding of the division operation.
• D. 36: This option is significantly larger and suggests a misunderstanding of the function's operations, perhaps involving multiplication instead of division and addition.

Analyzing incorrect options is a valuable learning exercise. By understanding the mistakes that lead to incorrect answers, we can strengthen our grasp of the concepts and avoid similar errors in the future. In this case, examining why options A, B, and D are incorrect helps to highlight the importance of following the order of operations and correctly interpreting the function's formula. This analytical approach not only improves problem-solving skills but also fosters a deeper understanding of the underlying mathematical principles.

Real-World Applications of Function Evaluation

Function evaluation isn't just a theoretical exercise; it has numerous real-world applications. For example:

• Physics: Calculating the position of an object at a specific time using a position function.
• Engineering: Determining the output of a system given a certain input.
• Economics: Predicting the demand for a product based on its price.
• Computer Science: Evaluating algorithms with different input sizes.

These examples illustrate the versatility and importance of function evaluation in various fields. Functions are used to model relationships between different quantities, and evaluating these functions allows us to make predictions, analyze data, and solve practical problems. Whether it's determining the trajectory of a projectile, designing an efficient algorithm, or forecasting economic trends, function evaluation plays a crucial role. Understanding how to work with functions is therefore an essential skill for anyone pursuing a career in STEM fields or any field that involves quantitative analysis.

Tips for Mastering Function Evaluation

Here are some tips to help you master function evaluation:

• Understand the notation: Make sure you understand what f(x) means and how it represents a function.
• Follow the order of operations: Always adhere to PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).
• Practice regularly: The more you practice, the more comfortable you'll become with evaluating functions.
• Break down complex problems: Divide complex functions into smaller steps to avoid errors.
• Check your work: Always double-check your calculations to ensure accuracy.

These tips provide a practical guide to improving your function evaluation skills. Understanding the notation and following the order of operations are fundamental. Regular practice is key to building fluency and confidence. Breaking down complex problems into smaller steps makes the process more manageable and reduces the risk of errors. Finally, checking your work is essential for ensuring accuracy and reinforcing your understanding. By incorporating these tips into your study habits, you can develop a strong foundation in function evaluation and excel in your mathematical pursuits.

Conclusion

In conclusion, evaluating f(x) = x/2 + 8 when x = 10 involves substituting 10 for x in the function's formula and simplifying the expression. The correct answer is 13. Understanding function evaluation is a fundamental skill in mathematics with wide-ranging applications. By mastering this concept, you'll be well-equipped to tackle more complex mathematical problems and real-world challenges. This process highlights the importance of careful substitution, following the order of operations, and paying attention to detail. Function evaluation is not just a mathematical exercise; it's a tool for understanding and modeling relationships between variables. By developing a strong understanding of functions and their evaluation, you can unlock a powerful set of tools for problem-solving and analysis in various disciplines.