Solving For F(2) Given F(x) = 5x - 12 A Step-by-Step Guide

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Introduction

In the realm of mathematics, functions serve as fundamental building blocks, establishing a clear relationship between inputs and their corresponding outputs. This article delves into the process of evaluating functions, using the specific example of f(x) = 5x - 12. We aim to find the value of the function when the input, x, is equal to 2, denoted as f(2). Understanding function evaluation is crucial for various mathematical concepts and applications, making it a cornerstone of mathematical literacy. This problem, a classic example of function evaluation, provides a solid foundation for grasping more complex mathematical concepts later on. This detailed explanation will walk you through the steps involved in solving this problem, ensuring a clear understanding of the underlying principles.

Decoding Function Notation

Before diving into the calculation, it's essential to understand the notation used to represent functions. The expression f(x) = 5x - 12 defines a function named "f". The x within the parentheses represents the input variable, and the entire expression 5x - 12 describes the rule that the function applies to this input. In essence, the function "f" takes an input x, multiplies it by 5, and then subtracts 12 from the result. This process highlights the core concept of a function: a mapping from inputs to outputs based on a defined rule. Grasping this notation is paramount for confidently navigating the world of functions and their applications. Think of a function as a mathematical machine: you put in a number (x), the machine performs some operations on it (in this case, multiplying by 5 and subtracting 12), and then it spits out a new number (f(x)).

The Essence of Function Evaluation

The core of the problem lies in function evaluation. This process involves substituting a specific value for the input variable in the function's expression and then simplifying the resulting expression. In our case, we are asked to find f(2), which means we need to replace every instance of x in the expression 5x - 12 with the number 2. This substitution transforms the function's general rule into a specific calculation, allowing us to determine the output corresponding to the input of 2. Mastering function evaluation is crucial for understanding how functions behave and for solving a wide range of mathematical problems. Function evaluation isn't just a mathematical exercise; it's a tool used in real-world applications, such as modeling physical phenomena or predicting trends. For instance, if f(x) represented the distance a car travels in x hours, then f(2) would tell us the distance the car travels in 2 hours.

Step-by-Step Solution: Calculating f(2)

Now, let's walk through the calculation step-by-step:

  1. Begin by writing down the function : f(x) = 5x - 12. This clearly establishes the function we are working with.
  2. Substitute x with 2: f(2) = 5(2) - 12. This is the crucial step where we replace the variable with the given input.
  3. Perform the multiplication : f(2) = 10 - 12. Following the order of operations, we multiply 5 by 2.
  4. Finally, perform the subtraction : f(2) = -2. This completes the calculation, giving us the output of the function when the input is 2.

Therefore, f(2) = -2. This meticulous step-by-step approach ensures accuracy and clarity, especially when dealing with more complex functions.

Identifying the Correct Answer

Looking at the given options:

A. -2 B. -22 C. 2 D. 10

We can clearly see that option A, -2, matches our calculated result. Thus, option A is the correct answer. This step reinforces the importance of careful calculation and comparison with the available options.

Common Pitfalls to Avoid

When evaluating functions, there are a few common errors that students often make. Being aware of these pitfalls can help you avoid them:

  • Incorrect Substitution: Ensure you replace every instance of the variable x with the given value. Forgetting to substitute in one place can lead to an incorrect answer.
  • Order of Operations: Always follow the order of operations (PEMDAS/BODMAS) when simplifying the expression after substitution. Failing to do so can result in a wrong calculation.
  • Sign Errors: Pay close attention to the signs (positive and negative) throughout the calculation. A small sign error can significantly alter the result.
  • Misinterpreting Notation: Understanding the function notation is crucial. Make sure you correctly interpret what the function is asking you to do.

By being mindful of these common pitfalls, you can significantly improve your accuracy in function evaluation.

Practice Problems for Mastery

To solidify your understanding of function evaluation, try these practice problems:

  1. If g(x) = 3x + 7, find g(3).
  2. Given h(x) = x² - 4, what is h(-2)?
  3. Let k(x) = (x + 1) / 2, determine k(5).

Working through these problems will help you build confidence and proficiency in evaluating functions. Remember, practice makes perfect! These practice problems offer varied scenarios, including different operations and negative inputs, to broaden your understanding.

Real-World Applications of Functions

Functions are not just abstract mathematical concepts; they have numerous real-world applications. Here are a few examples:

  • Physics: Functions can describe the motion of objects, the relationship between force and acceleration, and many other physical phenomena.
  • Economics: Supply and demand curves are functions that relate the price of a product to the quantity supplied or demanded.
  • Computer Science: Functions are the building blocks of computer programs, allowing programmers to write reusable code.
  • Finance: Compound interest calculations are based on exponential functions.

Understanding functions allows you to model and analyze these real-world situations mathematically. For instance, in computer science, a function might take a user's input (like a search query) and return relevant results. In finance, a function could calculate the future value of an investment based on the initial amount, interest rate, and time period.

Conclusion: The Power of Function Evaluation

In conclusion, evaluating functions is a fundamental skill in mathematics. By understanding the notation, the process of substitution, and the importance of the order of operations, you can confidently solve problems like finding f(2) when f(x) = 5x - 12. This skill serves as a stepping stone to more advanced mathematical concepts and has wide-ranging applications in various fields. So, embrace the power of function evaluation and continue exploring the fascinating world of mathematics! Mastering function evaluation opens doors to a deeper understanding of mathematical modeling and problem-solving. It empowers you to analyze relationships between quantities and make predictions based on mathematical rules. The ability to evaluate functions is a valuable asset in academic pursuits and beyond.