Finding (g-f)(n) With Given Functions: A Step-by-Step Guide

Nov 7, 2025 by Admin 60 views

Hey guys! Today, we're diving into a fun math problem where we'll be working with functions. Specifically, we're going to figure out how to find $(g-f)(n)$ when we're given two functions: $g(n) = 3n - 2$ and $f(n) = n^2 - 2$ . Don't worry, it's not as scary as it looks! We'll break it down step by step so you can ace this type of problem. So, grab your pencils and let's get started!

Understanding the Basics of Functions

Before we jump into solving for $(g-f)(n)$ , let's quickly recap what functions are all about. Think of a function like a machine: you feed it an input (in this case, 'n'), and it spits out an output based on a specific rule. In our case, the function $g(n)$ takes 'n', multiplies it by 3, and then subtracts 2. The function $f(n)$ takes 'n', squares it, and then subtracts 2. Functions are a fundamental concept in mathematics, and understanding them is crucial for tackling more complex problems. They allow us to model relationships between different variables and make predictions based on these relationships. The notation $g(n)$ and $f(n)$ represents the output of the function when the input is 'n'. For example, if we were to find $g(2)$ , we would substitute '2' for 'n' in the expression $3n - 2$ , resulting in $g(2) = 3(2) - 2 = 4$ . Similarly, to find $f(3)$ , we would substitute '3' for 'n' in the expression $n^2 - 2$ , giving us $f(3) = (3)^2 - 2 = 7$ . This process of substitution is key to evaluating functions and understanding their behavior. Remember, the variable 'n' is just a placeholder; we can use any variable or even an expression as the input to the function. This flexibility is one of the reasons why functions are such a powerful tool in mathematics and its applications. Understanding the input-output relationship of functions is essential for grasping more advanced concepts like function composition and inverse functions. So, make sure you're comfortable with the basics before moving on to more complex topics.

Breaking Down (g-f)(n)

Okay, so what does $(g-f)(n)$ actually mean? It's simply asking us to subtract the function $f(n)$ from the function $g(n)$ . Think of it like this: we're taking the output of $g(n)$ and subtracting the output of $f(n)$ for the same input 'n'. To do this, we'll first write out the expressions for both functions and then perform the subtraction. It's super important to pay attention to the order of operations and the signs, especially when dealing with negative numbers. This is a common area where mistakes can happen, so let's make sure we're extra careful. We'll also need to remember our algebra rules for combining like terms. For example, we can combine terms with the same variable and exponent, but we can't combine terms with different exponents. This is a fundamental concept in algebra, and it's essential for simplifying expressions correctly. By understanding the meaning of $(g-f)(n)$ and carefully following the steps of subtraction and simplification, we can confidently solve this type of problem. So, let's put on our thinking caps and get ready to dive into the calculations!

Step-by-Step Solution to Find (g-f)(n)

Alright, let's get our hands dirty and actually solve for $(g-f)(n)$ ! Here's how we'll do it, step by step:

Write out the functions: We know that $g(n) = 3n - 2$ and $f(n) = n^2 - 2$ . Let's write these down clearly so we don't mix them up.
Set up the subtraction: $(g-f)(n)$ means $g(n) - f(n)$ . So, we'll write that out: $(3n - 2) - (n^2 - 2)$ . Notice the parentheses! They're super important because we need to subtract the entire function $f(n)$ , not just the first term.
Distribute the negative sign: This is where things can get a little tricky if we're not careful. We need to distribute the negative sign in front of the parentheses to both terms inside. So, $(3n - 2) - (n^2 - 2)$ becomes $3n - 2 - n^2 + 2$ . Remember, subtracting a negative is the same as adding a positive!
Combine like terms: Now we'll look for terms that have the same variable and exponent. In this case, we have a constant term (-2) and its opposite (+2), which cancel each other out. So, we're left with $3n - n^2$ .
Rearrange for standard form (optional but recommended): It's common practice to write polynomials in descending order of exponents. So, we can rewrite $3n - n^2$ as $-n^2 + 3n$ . This just makes it look a little neater and easier to compare to answer choices.

And there you have it! We found that $(g-f)(n) = -n^2 + 3n$ .

Identifying the Correct Answer

Now that we've solved for $(g-f)(n)$ , let's take a look at the answer choices provided in the original problem: A. $n^2 - 7n - 5$ , B. $-n^2 + 3n$ , C. $-5n - 4$ , D. $n^2 - 3n - 4$ . By comparing our solution, $-n^2 + 3n$ , with the answer choices, we can clearly see that option B is the correct answer. This step is crucial to ensure we haven't made any calculation errors and that we're selecting the appropriate answer. Always double-check your work and compare your solution to the provided options to avoid careless mistakes. Sometimes, answer choices might look similar, so pay close attention to the signs and coefficients. Eliminating incorrect options can also be a helpful strategy if you're unsure of the correct answer. By carefully reviewing the answer choices and comparing them to your solution, you can increase your confidence in selecting the correct answer.

Common Mistakes to Avoid

Let's chat about some common pitfalls people often stumble into when tackling problems like this. Knowing these mistakes beforehand can help you dodge them and boost your accuracy!

Forgetting the parentheses: This is a biggie! When subtracting an entire function, you must use parentheses. Otherwise, you might only subtract the first term of the function and mess up the whole problem. Remember, the parentheses tell us to subtract the entire expression, not just a part of it.
Distributing the negative sign incorrectly: This goes hand-in-hand with the previous mistake. Make sure you distribute the negative sign to every term inside the parentheses. This means changing the sign of each term within the parentheses. A simple way to double-check is to imagine multiplying each term inside the parentheses by -1.
Combining unlike terms: You can only combine terms that have the same variable and exponent. For example, you can combine $3n$ and $-5n$ , but you can't combine $3n$ and $3n^2$ . It's like trying to add apples and oranges – they're different!
Rearranging terms incorrectly: When rearranging terms, make sure you keep the correct sign with each term. For example, if you have $-n^2 + 3n$ , the negative sign belongs to the $n^2$ term. Don't drop the negative sign!

By keeping these common mistakes in mind, you can significantly improve your chances of getting the correct answer. Remember to double-check your work, especially when dealing with negative signs and distribution.

Practice Problems for Extra Credit

Okay, guys, time to put your newfound skills to the test! Here are a couple of practice problems to help you solidify your understanding of finding $(g-f)(n)$ . Working through these will give you extra confidence and make you a function-subtracting pro!

Let $g(n) = 5n + 1$ and $f(n) = 2n^2 - 3$ . Find $(g-f)(n)$ .
If $g(n) = n^2 + 4n$ and $f(n) = -n + 2$ , what is $(g-f)(n)$ ?

Try solving these on your own, and don't peek at the solution until you've given it your best shot! Remember to follow the steps we discussed: write out the functions, set up the subtraction with parentheses, distribute the negative sign, combine like terms, and rearrange if needed. The more you practice, the more comfortable you'll become with these types of problems.

Conclusion: You've Got This!

Awesome job, guys! You've successfully learned how to find $(g-f)(n)$ when given two functions. Remember, the key is to understand the basics, pay attention to detail, and practice, practice, practice! By following the steps we've outlined and avoiding those common mistakes, you'll be well on your way to mastering function operations. Keep up the great work, and don't be afraid to tackle even more challenging problems. You've got this!