Solving F(x) = 2x – 5 When X = 3: A Math Problem

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Hey guys! Today, we're diving into a super common type of math problem that you'll likely encounter in algebra: evaluating a function. Specifically, we're tackling the question: What is the value of f(x) = 2x – 5 when x = 3? This might seem a bit daunting at first, but trust me, it's much simpler than it looks. We'll break it down step by step, making sure you understand not just the answer, but also the why behind it. So, grab your thinking caps, and let's get started!

The core concept here is function evaluation. In mathematical terms, a function, often denoted as f(x), is like a machine. You input a value (in this case, 'x'), and the machine performs a specific set of operations on that value, giving you an output. The equation f(x) = 2x – 5 is the rule that our "machine" follows. It tells us that for any input 'x', we should first multiply it by 2, and then subtract 5 from the result. The question is asking us: what happens when we feed '3' into this machine? To solve this, we simply substitute '3' for every 'x' we see in the equation. This is where the magic happens, and we transform the symbolic representation into a concrete calculation. We replace the abstract 'x' with the specific value '3', setting the stage for a straightforward arithmetic problem. It's all about understanding the notation and applying the given rule to the input value. Once you grasp this fundamental principle, you'll be able to tackle a wide range of function evaluation problems with confidence.

Step-by-Step Solution

Let's break down the solution into easy-to-follow steps:

  1. Understand the Function:

    • We are given the function f(x) = 2x – 5. This means that for any value of x, we multiply it by 2 and then subtract 5.

  2. Substitute the Value:

    • We are asked to find f(3), so we substitute x = 3 into the function:

      • f(3) = 2(3) – 5

    • This step is crucial. We're replacing the variable 'x' with the specific value we're interested in. Think of it like plugging a number into a formula.

  3. Perform the Calculation:

    • Now, we follow the order of operations (PEMDAS/BODMAS) to simplify the expression:

      • First, multiplication: 2 * 3 = 6

      • Then, subtraction: 6 – 5 = 1

  4. The Answer:

    • Therefore, f(3) = 1

    • So, the correct answer is A) 1.

This step-by-step approach highlights the importance of careful substitution and following the correct order of operations. Each step builds upon the previous one, leading us to the final solution. The beauty of mathematics lies in this logical progression, where each operation has a specific purpose and contributes to the overall result. By breaking down the problem into smaller, manageable steps, we make it less intimidating and more accessible. This method not only helps us arrive at the correct answer but also reinforces our understanding of the underlying mathematical principles.

Why This Matters

Why is evaluating functions so important? Well, functions are the building blocks of many mathematical models and real-world applications. They allow us to describe relationships between variables. For example:

  • In physics, a function might describe the position of an object over time.
  • In economics, a function could represent the cost of producing a certain number of items.
  • In computer science, functions are used extensively in programming to perform specific tasks.

Understanding how to evaluate functions is a fundamental skill that will help you in many different areas, both in math and beyond. It's not just about plugging in numbers; it's about understanding how quantities relate to each other. The ability to interpret and manipulate functions is crucial for problem-solving in various fields. This skill allows us to predict outcomes, analyze trends, and make informed decisions based on mathematical models. Furthermore, a strong grasp of functions lays the groundwork for more advanced mathematical concepts, such as calculus and differential equations. These concepts are used in engineering, finance, and many other disciplines to model complex systems and solve intricate problems. So, mastering function evaluation is a key step in building a solid foundation for future learning and career opportunities.

Common Mistakes to Avoid

  • Incorrect Order of Operations: Always remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Make sure you perform the multiplication before the subtraction in this case.
  • Sign Errors: Be careful with negative signs! A small mistake can lead to a completely wrong answer.
  • Substitution Errors: Double-check that you've substituted the value correctly for every instance of 'x' in the function.

These common pitfalls can easily trip up even the most careful students. A simple oversight in the order of operations can lead to a cascading series of errors, resulting in an incorrect final answer. Similarly, a misplaced negative sign can flip the entire outcome, highlighting the importance of meticulous attention to detail. Substitution errors, where the value is not correctly inserted for every instance of the variable, can also lead to significant deviations. The key to avoiding these mistakes is to cultivate a methodical approach. Double-check each step, paying close attention to the signs and the order of operations. It's also helpful to rewrite the function with the value substituted, clearly showing the operations to be performed. Practice and patience are essential in mastering these skills and developing the habit of error prevention.

Practice Makes Perfect

The best way to get comfortable with evaluating functions is to practice! Try these similar problems:

  1. What is the value of g(x) = 3x + 2 when x = -1?
  2. If h(x) = x² – 4, find h(2).
  3. Calculate p(x) = -2x + 7 when x = 0.

Working through these examples will help solidify your understanding of the process. Remember, math isn't a spectator sport; it's something you learn by doing. Each problem you solve is a step towards mastery. The more you practice, the more comfortable and confident you'll become in your ability to tackle different types of function evaluation questions. Don't be afraid to make mistakes; they're valuable learning opportunities. Analyze your errors, understand where you went wrong, and try again. With consistent effort and dedication, you'll find yourself becoming more proficient and efficient in solving these types of problems. The key is to stay engaged, keep practicing, and enjoy the process of learning and discovery.

Wrapping Up

So, there you have it! Evaluating functions is a fundamental skill in math. By understanding the concept of substitution and following the order of operations, you can easily solve these types of problems. Remember, practice is key, so keep working at it, and you'll be a pro in no time!

I hope this explanation has been helpful. Keep practicing, and you'll master function evaluation in no time! Remember, math is a journey, not a destination. Enjoy the process of learning, and don't be afraid to ask questions. You've got this!