Finding F(2) For F(x) = -x + 3: A Simple Guide

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Hey guys! Let's dive into a super straightforward math problem today. We're given a function, f(x) = -x + 3, and our mission is to figure out what f(2) is. Don't worry, it's much simpler than it sounds. Think of functions as little machines: you feed them a number (in this case, 2), and they spit out another number based on a specific rule. This rule is defined by the equation of the function. So, let’s break this down step by step and make sure we really understand what’s going on here. It’s not just about getting the right answer; it’s about grasping the concept behind it. Stick with me, and by the end, you’ll be solving these like a pro!

Understanding Functions

Before we jump into solving for f(2), let’s quickly recap what a function actually is. In simple terms, a function is like a mathematical machine. You input a value (often called 'x'), and the function performs some operations on it and then outputs a new value (often called 'y' or 'f(x)'). The equation f(x) = -x + 3 tells us exactly what this "machine" does: it takes your input 'x', multiplies it by -1, and then adds 3. This might seem a bit abstract now, but it's a fundamental concept in algebra and calculus. Really understanding functions opens the door to so many cool mathematical ideas, so it’s worth taking the time to get comfortable with them. We use functions all the time in the real world too, even if we don't realize it! Think about a vending machine – you put in money (the input), and it gives you a snack (the output). That’s kind of like a function in action!

Why is this important? Well, functions are the building blocks of more advanced math topics. They help us model relationships between different things, predict outcomes, and solve complex problems. Once you’ve got a solid grasp of functions, you can start exploring things like graphs, transformations, and even calculus. So, don't underestimate the power of understanding functions – they're a key skill for anyone who wants to excel in math and related fields. And honestly, they’re kind of fun once you get the hang of them. It’s like learning a secret code that unlocks a whole new way of thinking about the world.

Evaluating f(2)

Okay, now let's get our hands dirty and actually solve for f(2). Remember, f(x) = -x + 3. To find f(2), all we need to do is replace every 'x' in the equation with the number 2. This is called substitution, and it's a super important technique in algebra. So, we rewrite the equation as f(2) = -(2) + 3. See what we did there? We just swapped the 'x' for a '2'. Now it's just a matter of doing the arithmetic. First, we deal with the negative sign: -(2) is simply -2. So, our equation now looks like this: f(2) = -2 + 3. And finally, we add -2 and 3, which gives us 1. Therefore, f(2) = 1. Ta-da! We've found our answer. It really is that simple. The key is to take it one step at a time and not get intimidated by the notation.

Let's recap the steps just to make sure we've got it down: 1) Write down the function: f(x) = -x + 3. 2) Substitute 'x' with 2: f(2) = -(2) + 3. 3) Simplify: f(2) = -2 + 3. 4) Calculate: f(2) = 1. And that's it! We’ve successfully evaluated the function at x = 2. This process of substitution and simplification is something you’ll use again and again in math, so it’s worth practicing until it feels completely natural.

Step-by-Step Solution

Let's break down the solution into very clear, easy-to-follow steps. This will help solidify your understanding and make sure you can tackle similar problems with confidence. Here’s the breakdown:

  1. Write down the function: Our starting point is the given function, f(x) = -x + 3. This is the rule that tells us what to do with our input. Think of it as the recipe for our mathematical machine.
  2. Substitute x with 2: This is where the magic happens. We replace every 'x' in the equation with the value we're interested in, which is 2. So, f(x) becomes f(2), and -x + 3 becomes -(2) + 3. It’s like swapping out a variable for a specific number.
  3. Simplify: Now we need to tidy things up. The expression -(2) is the same as -2. So, we rewrite our equation as f(2) = -2 + 3.
  4. Calculate: The final step is to perform the addition. -2 + 3 equals 1. Therefore, f(2) = 1. We've arrived at our answer!

Each of these steps is crucial. Skipping one or doing them out of order can lead to mistakes. So, always take your time and work through each step carefully. With practice, this process will become second nature, and you'll be able to solve these kinds of problems in your sleep! Remember, math is like building with blocks – you need to have a solid foundation in the basics before you can move on to more complex ideas.

Another Example

To make sure we've really nailed this, let's try another example. Suppose we have the function g(x) = 2x - 1, and we want to find g(4). What do we do? Well, we follow the same steps as before. First, we write down the function: g(x) = 2x - 1. Next, we substitute 'x' with 4: g(4) = 2(4) - 1. Now we simplify: g(4) = 8 - 1. And finally, we calculate: g(4) = 7. See? It's the same process, just with a different function and a different input value.

Let's walk through one more, just for good measure. This time, let’s say we have h(x) = x² + 2, and we want to find h(-3). Don't let the x² scare you – it just means we need to square the input value. So, we write down the function: h(x) = x² + 2. We substitute 'x' with -3: h(-3) = (-3)² + 2. Now we simplify: h(-3) = 9 + 2. And finally, we calculate: h(-3) = 11. The key takeaway here is that the process is always the same, regardless of the complexity of the function. Just remember to substitute carefully, simplify step by step, and don't be afraid to practice!

Conclusion

So, there you have it! Finding f(2) for f(x) = -x + 3 is as simple as substituting and calculating. The answer is f(2) = 1. But more importantly, we've walked through the process step by step, reinforcing the fundamental concept of function evaluation. Remember, functions are the workhorses of mathematics, and understanding how they work is crucial for success in algebra and beyond. Keep practicing, and you'll become a function-evaluating whiz in no time! Don't hesitate to try out different functions and different input values to really solidify your understanding. And if you ever get stuck, remember to break the problem down into smaller, manageable steps. Math is all about building skills and confidence, one step at a time.