Find (f-g)(x) For F(x)=2x+1 And G(x)=x^2-7

Hey guys! Let's dive into a fun mathematical problem today. We're given two functions, f(x) = 2x + 1 and g(x) = x^2 - 7, and our mission is to find the expression for (f-g)(x). Sounds like a puzzle, right? Well, let’s break it down step by step so it's super easy to understand. This kind of problem is super common in algebra, so mastering it now will definitely pay off later. We will go through the basic definitions, the steps to solve, and some common mistakes to avoid. So grab your calculators and let's get started!

Understanding Function Operations

Before we jump into the problem, let's quickly refresh our understanding of function operations. When we talk about (f-g)(x), we're essentially talking about subtracting the function g(x) from the function f(x). It’s like saying, “Hey, for any value of x, what’s the result of f(x) minus g(x)?” This is a fundamental concept in algebra and calculus, so nailing this down is crucial. Function operations aren’t limited to just subtraction; we can also add, multiply, and divide functions. Each operation has its own quirks, but the basic idea remains the same: we’re performing an arithmetic operation on the outputs of the functions for a given input x. The notation might look a bit intimidating at first, but trust me, once you get the hang of it, it's smooth sailing. Think of functions as little machines that take an input, do some calculations, and spit out an output. When we perform operations on functions, we’re essentially combining these machines in different ways. So, with that in mind, let's tackle our specific problem.

Step-by-Step Solution for (f-g)(x)

Alright, let's get our hands dirty and solve this thing! Here’s how we find (f-g)(x) when f(x) = 2x + 1 and g(x) = x^2 - 7:

Step 1: Write down the definition

First things first, let's write down what (f-g)(x) actually means. By definition:

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

This is our starting point, our foundation. It’s like the recipe for our mathematical dish. Without this, we're just throwing ingredients together without a plan. So, always start by writing down the definition. It helps keep things clear and prevents confusion down the road.

Step 2: Substitute the functions

Now comes the fun part – substitution! We know what f(x) and g(x) are, so let’s plug them into our definition:

(f-g)(x) = (2x + 1) - (x^2 - 7)

See? We're just replacing the function symbols with their actual expressions. It’s like swapping out the names for the actual ingredients in our recipe. Make sure you use parentheses here! They’re super important because they tell us to treat the entire expression of g(x) as a single unit when we subtract. Forgetting the parentheses is a common mistake, and it can totally throw off your answer. So, parentheses are your friends in this step!

Step 3: Distribute the negative sign

This is where things can get a little tricky, but don’t worry, we’ve got this! We need to distribute the negative sign in front of the parentheses to each term inside:

(f-g)(x) = 2x + 1 - x^2 + 7

Notice how the - sign in front of (x^2 - 7) changes the signs inside the parentheses? The x^2 becomes -x^2, and the -7 becomes +7. This is a crucial step, and it’s where a lot of mistakes happen. So, take your time and double-check that you’ve distributed the negative sign correctly. It’s like making sure you’ve mixed all the ingredients thoroughly – you don’t want any clumps!

Step 4: Combine like terms

Almost there! Now we need to combine the like terms. Like terms are those that have the same variable raised to the same power. In our case, we have a constant term 1 and another constant term 7, which we can combine:

(f-g)(x) = -x^2 + 2x + 1 + 7

(f-g)(x) = -x^2 + 2x + 8

We’ve combined the 1 and 7 to get 8. And that’s it! We’ve simplified the expression as much as we can. It’s like the final touch on our dish, making sure everything is perfectly balanced.

The Final Answer

So, after all that awesome math work, we've found that:

(f-g)(x) = -x^2 + 2x + 8

That means the correct answer is D. $-x^2+2x+8$. Pat yourself on the back, guys! You've successfully navigated a function operation problem. This is a big win! Remember, math is like building with blocks. Each step builds on the previous one, and once you understand the basics, you can tackle more complex problems with confidence. So, keep practicing, keep asking questions, and keep building those math skills!

Common Mistakes to Avoid

Alright, let’s chat about some common pitfalls that students often encounter when solving problems like this. Knowing these mistakes can help you dodge them and ace your math problems like a pro!

Forgetting the Parentheses

This is a huge one! We talked about it earlier, but it’s worth repeating. When you substitute g(x) into (f-g)(x), you need to use parentheses around the entire expression for g(x). Forgetting the parentheses can lead to incorrect distribution of the negative sign, which will totally mess up your answer. Think of the parentheses as a protective shield around the function, ensuring that the negative sign affects everything inside it.

Incorrectly Distributing the Negative Sign

Another common mistake is messing up the distribution of the negative sign. Remember, the negative sign in front of the parentheses changes the sign of every term inside. So, if you have -(x^2 - 7), it becomes -x^2 + 7. Make sure you're careful and methodical when you distribute. It's like double-checking your packing list before a trip – you want to make sure you haven't forgotten anything important!

Combining Unlike Terms

This is a classic algebra blunder. You can only combine terms that have the same variable raised to the same power. For example, you can combine 2x and 3x because they both have x raised to the power of 1, but you can't combine 2x and 2x^2 because the powers of x are different. It’s like trying to mix oil and water – they just don’t blend! Always double-check that you’re only combining like terms.

Making Arithmetic Errors

Sometimes, the simplest arithmetic errors can trip you up. Adding or subtracting numbers incorrectly can lead to the wrong answer, even if you've done everything else right. So, take your time and double-check your calculations, especially when dealing with negative numbers. It’s like proofreading your essay – a fresh pair of eyes can catch mistakes you might have missed.

Practice Problems

Okay, now that we've tackled the main problem and discussed common mistakes, it's time for some practice! Here are a couple of problems for you to try on your own. Remember, practice makes perfect, so don't be afraid to dive in and give it your best shot.

Practice Problem 1

Let f(x) = 3x^2 + 2x - 1 and g(x) = x^2 - 5x + 4. Find (f-g)(x).

Practice Problem 2

Given f(x) = 4x - 3 and g(x) = 2x^2 + 1, determine the expression for (f-g)(x).

Go ahead and work through these problems. Use the steps we discussed earlier, and keep an eye out for those common mistakes. The more you practice, the more comfortable you'll become with these types of problems. And remember, if you get stuck, don't hesitate to review the steps or ask for help. We’re all in this together!

Conclusion

So, guys, we've journeyed through the process of finding (f-g)(x) for given functions. We've broken down the steps, highlighted common mistakes, and even given you some practice problems to flex those math muscles. Remember, the key to mastering these concepts is understanding the fundamentals and practicing consistently. Math can be challenging, but it’s also super rewarding when you see the pieces come together. Keep up the awesome work, and I’ll catch you in the next math adventure! Keep practicing, and you'll be a function operation whiz in no time! You've got this!