Solving $(g-f)(3)$: A Step-by-Step Guide

Hey Plastik Magazine readers! Let's dive into a fun math problem together. Today, we're tackling the question: If $f(x) = 4 - x^2$ and $g(x) = 6x$, which expression is equivalent to $(g - f)(3)$? Sounds a bit intimidating at first, right? But trust me, we'll break it down into easy-to-understand steps. Get ready to flex those math muscles and feel super smart! This isn't just about finding the right answer; it's about understanding why that answer is correct. So, let's get started, guys!

Understanding the Problem: What Does $(g - f)(3)$ Mean?

Alright, before we jump into the calculations, let's make sure we're all on the same page. The expression $(g - f)(3)$ might look a bit like math jargon, but it's actually pretty straightforward once you break it down. Essentially, it means we need to do two things: First, subtract the function f(x) from the function g(x). Second, we need to evaluate this new function at x = 3. Think of it like a recipe. You have two ingredients, f and g, and the instructions tell you to combine them in a specific way, and then use a specific amount of each. In this case, our ingredients are the functions f(x) = 4 - x^2 and g(x) = 6x. The instructions tell us to subtract f(x) from g(x) and then plug in 3 wherever we see x.

So, when we see $(g - f)(3)$, we're essentially asking: "What do we get when we subtract the value of f at x = 3 from the value of g at x = 3?" It's all about substituting the value of x in each function and then performing the subtraction. We're not just dealing with numbers here; we're dealing with functions – mathematical machines that take an input (x) and give you an output based on a specific set of rules. Understanding this concept is super important because it forms the bedrock for more advanced topics in math like calculus and differential equations. Getting comfortable with function notation now will seriously pay off later, believe me. So, let's get our hands dirty and see how we solve this step by step. We'll find out which of the multiple-choice options, A, B, C, or D, is equivalent to our answer, and we'll have a blast doing it!

Step-by-Step Solution: Finding the Value of $(g - f)(3)$

Now, let's get to the fun part: solving the problem step by step. This is where we put our understanding to the test and calculate the value of $(g - f)(3)$. Remember, we have f(x) = 4 - x^2 and g(x) = 6x. Let's tackle this methodically, shall we?

1. Find g(3): First, we need to find the value of g(3). Since g(x) = 6x, we substitute x with 3: g(3) = 6 * 3 = 18. Easy peasy, right?
2. Find f(3): Next, we need to find the value of f(3). Since f(x) = 4 - x^2, we substitute x with 3: f(3) = 4 - 3^2 = 4 - 9 = -5. We've found the values of both functions at x = 3!
3. Calculate (g - f)(3): Finally, we subtract f(3) from g(3): (g - f)(3) = g(3) - f(3) = 18 - (-5) = 18 + 5 = 23. Bam! There you have it. The value of $(g - f)(3)$ is 23. This is how we take the abstract concept and turn it into a concrete numerical result. Each step builds on the previous one, and before you know it, you've solved the problem. The beauty of mathematics lies in its logical progression. You start with a set of rules, apply them systematically, and arrive at a definitive answer. No guesswork involved. Just clear, concise steps leading to the solution. Now that we have our answer (23), we need to see which of the provided options matches it. We know the answer, so let's see which expression is equivalent to this final result.

Matching the Solution with the Answer Choices

Okay, awesome! Now that we've found that $(g - f)(3) = 23$, we need to see which of the multiple-choice options gives us the same result. Let's go through them one by one, shall we? This is where we check our work and make sure our understanding of the problem aligns with the answer choices. Keep in mind that we're looking for an expression that, when evaluated, also gives us 23. So, let's break down each option and see if it's the correct one:

• Option A: 6 - 3 - (4 + 3)^2 Let's evaluate it: 6 - 3 - (7)^2 = 6 - 3 - 49 = 3 - 49 = -46. Definitely not 23. So, Option A is wrong.
• Option B: 6 - 3 - (4 - 3^2) Let's calculate: 6 - 3 - (4 - 9) = 6 - 3 - (-5) = 3 + 5 = 8. Nope, this is not the answer. Option B is incorrect.
• Option C: 6(3) - 4 + 3^2 Evaluating this: 18 - 4 + 9 = 14 + 9 = 23. Bingo! Option C is looking promising. Let's see if this one is the right answer.
• Option D: 6(3) - 4 - 3^2 Let's see if this option matches: 18 - 4 - 9 = 14 - 9 = 5. Not the answer, so Option D is wrong.

Therefore, the correct answer is C: 6(3) - 4 + 3^2. Because when we evaluate this expression, we get 23, the same as our earlier calculation of (g - f)(3). That's a wrap, guys! Not too tough, right? See, math can be enjoyable when you break it down into manageable chunks and check your work.

Conclusion: Mastering the Art of Function Evaluation

So, there you have it, Plastik Magazine readers! We've successfully solved the problem and identified the correct answer: Option C. What a journey, right? We started with a potentially intimidating question and broke it down into simple, easy-to-understand steps. Along the way, we learned about function notation, substitution, and the importance of following a logical process. The key takeaways from this exercise are crucial for building a strong foundation in mathematics. We practiced function evaluation, understanding that plugging in a value for x is a core concept. We also reinforced our algebra skills by meticulously calculating the values. And finally, we saw the significance of checking our work to ensure accuracy. Remember, the journey through mathematics is about more than just finding the correct answer; it's about developing critical thinking and problem-solving skills that apply far beyond the classroom. Mathematics is all about logic and how it can be applied to solve the real-world problems. Keep practicing, keep exploring, and most importantly, keep having fun with it! Keep an eye out for more math challenges from Plastik Magazine. Until next time, stay curious, stay engaged, and keep those brain cells active, you guys! We hope you enjoyed the article. Let us know if you want to see more math articles or other types of articles. We are here to help you!