Finding F(2) For F(x) = 2/x^2: A Step-by-Step Solution

Have you ever stumbled upon a mathematical function and wondered how to find its value at a specific point? It's a common question, especially in algebra and calculus. In this article, we'll break down the process of finding the value of a function for a given input, using the example f(x) = 2/x^2 when x = 2. Let's dive in and make math a little less mysterious!

Understanding the Function f(x) = 2/x^2

Before we jump into the calculation, let's make sure we understand what the function f(x) = 2/x^2 actually means. In simple terms, this function takes an input value (which we call 'x'), squares it, and then divides 2 by the result. Functions are like little machines: you feed them a number, and they spit out another number based on a specific rule. Identifying the rule is the first step in solving problems involving functions. For this function, the rule is quite straightforward: square the input, then divide 2 by the result. This type of function, where the variable is in the denominator, is known as a rational function. Rational functions can behave in interesting ways, especially when x gets very large or very small, or when x approaches zero. Understanding the behavior of functions like this is crucial in many areas of mathematics and its applications. Think about scenarios like calculating the gravitational force between two objects (which decreases with the square of the distance) or understanding the intensity of light as you move away from the source (which also follows an inverse square law). By grasping the fundamentals of functions, you're not just learning math for the sake of math; you're building a foundation for understanding the world around you.

The Task: Finding f(2)

Our specific task is to find the value of f(x) when x = 2. This means we need to substitute 2 for every 'x' we see in the function's equation. The notation f(2) is a concise way of saying "the value of the function f when x is 2." It's like a shorthand that mathematicians use to communicate ideas efficiently. This is a fundamental concept in evaluating functions, and it's a skill you'll use repeatedly in algebra, calculus, and beyond. Think of it like this: if f(x) is a recipe, then f(2) is what you get when you follow the recipe using 2 as your main ingredient. The process of substitution is the key to unlocking the value of the function at a specific point. Once you've mastered substitution, you'll be able to tackle a wide range of problems involving functions. The idea is to replace the general variable x with the specific value we're interested in, which in this case is 2. This simple substitution transforms the abstract function f(x) into a concrete calculation that we can easily perform. By carefully substituting and simplifying, we'll arrive at the numerical value of the function at that point. So, let's get started with the substitution and see what we find!

Step-by-Step Solution

Here's how we solve it:

1. Substitute x with 2: We replace x with 2 in the function: f(2) = 2/(2^2)
2. Calculate 2 squared: 2 squared (2^2) is 2 * 2 = 4. So now we have: f(2) = 2/4
3. Simplify the fraction: 2/4 can be simplified to 1/2, which is equal to 0.5. Therefore, f(2) = 0.5

That's it! We've found that the value of the function f(x) = 2/x^2 when x = 2 is 0.5. Each of these steps is crucial in arriving at the correct answer. The substitution step is where we bridge the gap between the general function and the specific value we want to find. The exponentiation step simplifies the expression, making it easier to work with. And the final simplification step gives us the answer in its simplest form. Understanding these individual steps and how they fit together is key to mastering function evaluation. It's not just about getting the right answer; it's about understanding the process and being able to apply it to other similar problems. By breaking down the problem into these manageable steps, we've made the solution process clear and easy to follow. Now you can confidently apply this same method to evaluate other functions at different points.

The Answer

The value of f(x) when x = 2 is 0.5. So the correct answer is D. 0.5. This straightforward calculation demonstrates the core concept of function evaluation. By substituting the given value into the function and simplifying, we can find the output that corresponds to that input. This process is fundamental to understanding how functions work and is used extensively in various mathematical and scientific fields. Think about how this concept applies in real-world scenarios. For example, if f(x) represented the distance traveled by a car after x hours, then f(2) would tell you how far the car has traveled after 2 hours. Or, if f(x) represented the cost of producing x items, then f(2) would tell you the cost of producing 2 items. The power of functions lies in their ability to model and predict relationships between different quantities. By understanding how to evaluate functions, you're gaining a powerful tool for analyzing and solving problems in a wide range of contexts. So, keep practicing, and you'll become a master of function evaluation!

Why This Matters

Understanding how to evaluate functions is a cornerstone of mathematics. It's a skill that you'll use in algebra, calculus, and beyond. Functions are used to model relationships between variables in almost every field, from physics and engineering to economics and computer science. For instance, in physics, you might use a function to describe the trajectory of a projectile. In economics, you might use a function to model the demand for a product as a function of its price. In computer science, functions are the building blocks of programs, allowing you to encapsulate reusable logic. Being able to evaluate functions allows you to make predictions, solve problems, and gain insights into these relationships. Mastering this skill not only helps you succeed in math courses but also opens doors to understanding and analyzing the world around you. Think of functions as a language for describing patterns and relationships. Just as learning a foreign language allows you to communicate with people from different cultures, learning functions allows you to communicate with the language of mathematics and science. The more fluent you become in this language, the more effectively you can understand and interact with the world.

Practice Makes Perfect

The best way to get comfortable with function evaluation is to practice. Try evaluating other functions for different values of x. You can start with simple linear functions like f(x) = 3x + 1 or quadratic functions like f(x) = x^2 - 2x + 3. Then, you can move on to more complex functions like rational functions or trigonometric functions. The key is to break down the problem into small steps, just like we did in this article. Start by substituting the value of x into the function, then simplify the expression step-by-step. Don't be afraid to make mistakes – they are a natural part of the learning process. The more you practice, the more confident you'll become in your ability to evaluate functions. And as you gain confidence, you'll start to see the connections between different mathematical concepts and how functions play a central role in those connections. So, grab a pencil and paper, find some functions to work with, and start practicing! The rewards of mastering function evaluation are well worth the effort.

Conclusion

We've successfully found the value of f(x) = 2/x^2 when x = 2. By understanding the function and following a step-by-step approach, we arrived at the answer: 0.5. Remember, the key to mastering function evaluation is understanding the process and practicing regularly. Keep exploring different functions and values, and you'll become a pro in no time!

For further learning about functions, you can explore resources like Khan Academy's Functions and Equations section. It's a great place to find more examples, practice problems, and explanations.