Finding (f - G)(x): A Step-by-Step Math Guide

Hey math enthusiasts! Let's dive into a common problem type: finding the difference of two functions. Specifically, we're going to figure out how to calculate $(f - g)(x)$ when given the functions $f(x)$ and $g(x)$. It's not as scary as it might sound, I promise! We'll break it down step by step, making sure you understand the core concepts. By the end of this, you'll be a pro at subtracting functions!

Understanding the Basics: Function Subtraction

First things first, let's get our fundamentals straight. When we see $(f - g)(x)$, what does it actually mean? Well, it's just a shorthand way of saying: "Subtract the function g(x) from the function f(x)." Literally, we're taking the output of f(x) and subtracting the output of g(x) for the same input value x. So, if we know what f(x) and g(x) are, we can simply substitute them into the expression and simplify. This operation is pretty straightforward, and once you get the hang of it, you'll find it's a piece of cake. The key is to keep track of the minus sign and distribute it correctly, as that's where most folks tend to slip up. Always remember the parentheses! They are your best friend here, helping you avoid common errors.

Now, let's talk about the specific functions we're dealing with in this problem. We are given $f(x) = 3x - 2$ and $g(x) = 2x + 1$. These are linear functions, meaning their graphs are straight lines. The goal is to find a new function, $(f - g)(x)$, that represents the difference between these two. To achieve this, you need to subtract the expression for g(x) from the expression for f(x). It's like having two recipes and finding out the difference in the ingredient amounts. Let's make sure we clearly understand the mechanics of function subtraction. For example, if $x$ were equal to 1, we would first find $f(1)$ and $g(1)$, and then we'd subtract $g(1)$ from $f(1)$. That gives us one point on the graph of the new function $(f - g)(x)$. The process is the same for every single value of x. The goal is to come up with a single simplified expression that will yield the result for any valid x value, thus describing the new function in a compact way. Don't worry, we'll go through the calculations step by step!

Step-by-Step Calculation: Finding (f - g)(x)

Alright, let's roll up our sleeves and get to work! We're going to take the functions given and go through the subtraction process. Remember, we are trying to find $(f - g)(x)$.

1. Write down the expression: Start by writing down the functions with a subtraction sign in between:

   $(f - g)(x) = f(x) - g(x) = (3x - 2) - (2x + 1)$.

   See how we've replaced f(x) and g(x) with their respective expressions? This is a crucial step to start with. Notice the parentheses around the expressions for $f(x)$ and $g(x)$. They are essential, especially when there's a minus sign in between. We're getting the equation ready to be simplified. Make sure to be meticulous and maintain these parentheses to avoid any sign errors during the subsequent steps. This organization is key to a smooth and accurate solution.

2. Distribute the negative sign: This is where things can get a little tricky, so pay close attention. The minus sign in front of the $(2x + 1)$ expression means we need to distribute it to both terms inside the parentheses. In essence, you are multiplying each term inside the parentheses by $-1$:

   $(3x - 2) - (2x + 1) = 3x - 2 - 2x - 1$.

   Distributing the minus sign flips the sign of each term inside the second set of parentheses. It changes the $+2x$ into $-2x$ and the $+1$ into $-1$. This distribution step is the most common place where errors occur. Always double-check that you've correctly applied the negative sign to every term within the parentheses. If you get this step wrong, the rest of your solution will be off. So take your time and do it right!

3. Combine like terms: Now, let's gather the 'like' terms (terms with the same variable and exponent) and combine them. Grouping the x terms together and the constant terms together, we get:

   $3x - 2x - 2 - 1$.

   Then simplify it to:

   $(3x - 2x) + (-2 - 1) = x - 3$.

   We combine the x terms: $3x - 2x = x$. And we combine the constant terms: $-2 - 1 = -3$.

   The result is:

   $(f - g)(x) = x - 3$.

   And there you have it! The simplified expression for $(f - g)(x)$ is $x - 3$. This is the final answer, and it represents a new linear function.

Choosing the Correct Answer: Matching the Solution

Now that we've found that $(f - g)(x) = x - 3$, let's go back to the multiple-choice options you provided. The correct answer should match our simplified expression. In this case, we're looking for the option that says $x - 3$. Looking at the answer choices you provided:

A. $3 - x$ B. $x - 3$ C. $5x - 1$ D. $5x - 3$

It's pretty clear that B. $x - 3$ is the correct answer. This is the only option that matches the result of our calculation. Always remember to double-check your answer with the given options to ensure you've selected the right one. You might have calculated the correct result, but made a small mistake selecting the answer, or misread the options provided. Take your time, and carefully match your result with the provided options. This is a very important step to getting the correct answer!

Tips for Success: Avoiding Common Mistakes

To make sure you nail these problems every time, let's go over a few key tips and tricks. These can help you avoid some of the most common pitfalls.

• Parentheses are your friends: Always use parentheses, especially when subtracting functions. They help you remember to distribute the negative sign correctly. Missing parentheses is a common source of error in these problems.
• Double-check distribution: The most common mistake is in distributing the negative sign. Carefully check that you've applied it to every term within the parentheses. Remember, every term gets multiplied by the negative one. This one step causes a lot of mistakes, so make sure you are super careful.
• Combine like terms: Be organized when combining like terms. Group the x terms together, and group the constant terms together. This makes it easier to keep track of everything and avoid errors. Organize and simplify as you go, and always keep your work neat. A clean paper will help you avoid careless mistakes.
• Practice, practice, practice: The more you practice these types of problems, the easier they will become. Work through different examples to get comfortable with the process. Practice helps you get faster, more accurate, and more confident in your abilities. Practice makes perfect, so be sure to practice.

Conclusion: Mastering Function Subtraction

So there you have it! We've successfully found $(f - g)(x)$ by subtracting g(x) from f(x). This is a fundamental concept in mathematics, and now you have the skills to tackle similar problems with confidence. Keep practicing, and you'll become a function subtraction master in no time! Remember to always stay organized, and to carefully follow the steps. And most of all, never be afraid to ask for help or go back and review any steps you're unsure of. You've got this! Good luck, and happy calculating!