Finding (f-g)(2) Given F(x) And G(x)

Hey guys! Today, we're diving into a fun math problem that involves finding the value of (f-g)(2) when we're given two functions, f(x) and g(x). It might sound a bit intimidating at first, but trust me, it's super manageable once we break it down step by step. So, let's jump right in and make sure we nail this concept. This exploration isn't just about solving this specific problem; it's about equipping you with the skills and confidence to tackle similar challenges in mathematics. Whether you're a student brushing up on your algebra or just a math enthusiast looking for a mental workout, you're in the right place. Let's make math enjoyable and understandable together!

Understanding the Functions

Before we can find (f-g)(2), we need to understand what the functions f(x) and g(x) actually do.

• f(x) = 3x² + 1: This function takes an input 'x', squares it, multiplies the result by 3, and then adds 1. So, if we input 2 into this function, we'll be doing 3 * (2²) + 1.
• g(x) = 1 - x: This function is simpler; it takes an input 'x' and subtracts it from 1. If we put 2 into this function, we get 1 - 2.

It's crucial to grasp what these functions do individually before we combine them. Think of them as little machines that perform specific operations on any number you feed them. Understanding these operations is the key to solving the problem at hand. When you encounter functions like these, take a moment to visualize the steps involved. This mental exercise not only clarifies the function's behavior but also helps in applying the function correctly in various mathematical contexts. Remember, functions are fundamental tools in mathematics, and mastering them opens doors to more complex and fascinating concepts.

Defining (f-g)(x)

Now that we understand f(x) and g(x) separately, let's figure out what (f-g)(x) means. The notation (f-g)(x) simply means we're subtracting the function g(x) from the function f(x). Basically, we're taking the entire expression for f(x) and subtracting the entire expression for g(x) from it. Mathematically, it looks like this:

(f-g)(x) = f(x) - g(x)

This is a key step because it sets up the equation we'll use to solve the problem. It's like having a recipe – you need to know which ingredients (functions) to combine and how (subtraction) before you can start cooking (solving). The concept of combining functions is prevalent in advanced math topics, so getting comfortable with this notation now will pay dividends later. Think of function combination as a mathematical recipe, where you're blending different operations to create something new. This perspective not only simplifies the process but also highlights the elegance and interconnectedness of mathematical concepts.

Calculating (f-g)(x)

Let's plug in the actual functions into our (f-g)(x) equation:

(f-g)(x) = (3x² + 1) - (1 - x)

Now, we need to simplify this expression. Remember to distribute the negative sign to both terms inside the parentheses of g(x):

(f-g)(x) = 3x² + 1 - 1 + x

Notice how the '1' and '-1' cancel each other out. This leaves us with:

(f-g)(x) = 3x² + x

We've now found the new function (f-g)(x)! This simplified expression is much easier to work with when we need to evaluate it at a specific point. Simplifying expressions is a fundamental skill in algebra, and it's like decluttering your workspace before starting a project. By removing unnecessary terms and combining like terms, you make the expression more manageable and less prone to errors. Mastering this skill not only makes problem-solving more efficient but also deepens your understanding of the underlying mathematical structure.

Evaluating (f-g)(2)

Okay, we're in the home stretch! We need to find the value of (f-g)(2). This means we need to substitute '2' for 'x' in our (f-g)(x) equation:

(f-g)(2) = 3(2²) + 2

Let's break this down:

• 2² = 4
• 3 * 4 = 12
• 12 + 2 = 14

So, (f-g)(2) = 14.

This final step is where all our previous work comes together. It's like the grand finale of a performance, where you showcase the skills you've honed throughout the process. Substituting a value into a function is a common operation in mathematics, and it's crucial for understanding how functions behave and for making predictions based on mathematical models. By carefully following the order of operations and paying attention to detail, you can confidently evaluate any function at any given point. This skill is not just about getting the right answer; it's about developing a deep understanding of the relationship between inputs and outputs in mathematical functions.

The Answer

The value of (f-g)(2) is 14. So, the correct answer is B. 14.

Why This Matters

This type of problem is more than just a math exercise. It demonstrates how we can combine functions, which is a powerful concept in mathematics and many real-world applications. For example, in physics, you might combine functions to model the motion of an object under the influence of multiple forces. In computer graphics, functions are used to create complex shapes and animations. Understanding function operations gives you a strong foundation for tackling more advanced topics in mathematics and beyond. It's like learning the alphabet before writing a novel; you need the basic building blocks to create something complex and meaningful. The ability to manipulate and combine functions is a key skill for anyone pursuing a career in science, technology, engineering, or mathematics, and it's also valuable in many other fields that rely on data analysis and problem-solving.

Practice Makes Perfect

The best way to get comfortable with these types of problems is to practice! Try working through similar examples with different functions. Experiment with adding, subtracting, multiplying, and even dividing functions. The more you practice, the more confident you'll become. And hey, don't be afraid to make mistakes – they're a natural part of the learning process. When you encounter a challenge, take it as an opportunity to deepen your understanding and refine your skills. Remember, math is not a spectator sport; it's something you learn by doing. So, grab a pencil, open your notebook, and start practicing! The more you engage with the material, the more you'll discover the beauty and power of mathematics.

Conclusion

So, there you have it! We've successfully navigated through finding (f-g)(2) by understanding the individual functions, combining them, and then evaluating the result. Remember, the key is to break down the problem into smaller, manageable steps. With practice and a solid understanding of the fundamentals, you can conquer any function-related challenge that comes your way. Keep up the great work, guys, and I'll catch you in the next math adventure! This journey through function operations underscores the importance of a structured approach to problem-solving. By breaking down a complex task into smaller, more manageable steps, we not only simplify the process but also enhance our understanding of the underlying concepts. This approach is not only valuable in mathematics but also in various aspects of life, from project management to personal goal setting. So, as you continue your mathematical journey, remember that a systematic and methodical approach is your best ally in conquering any challenge.