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

Hey math enthusiasts! Let's dive into a common algebra problem: figuring out the value of $(g-f)(3)$ when we're given the functions $f(x) = 4 - x^2$ and $g(x) = 6x$. This type of question often pops up in algebra, and understanding it is key to mastering function operations. Don't worry, it's not as scary as it looks. We'll break it down into easy-to-follow steps.

Understanding the Problem: (g-f)(3)

First things first, what does $(g-f)(3)$ even mean? In function notation, $(g-f)(3)$ represents the difference between the values of the functions $g(x)$ and $f(x)$ when $x = 3$. Essentially, we're going to:

1. Find the value of $g(3)$.
2. Find the value of $f(3)$.
3. Subtract the value of $f(3)$ from the value of $g(3)$.

It's like having two separate calculations and then putting them together. Think of it like this: If you have two recipes, one for a cake and one for frosting. $(g-f)(3)$ is like saying, "How much more frosting do I have than cake?" when I make 3 servings. Clear, right? We’re not actually baking, but we are doing similar operations.

Step-by-Step Solution: Finding g(3) and f(3)

Let’s start with finding $g(3)$. We know that $g(x) = 6x$. To find $g(3)$, we substitute $x$ with $3$ in the equation:

$g(3) = 6 * 3$ $g(3) = 18$

So, $g(3)$ equals 18. This is the first piece of our puzzle, and it's pretty straightforward. We’ve simply replaced $x$ with a numerical value and performed a basic multiplication. Not too hard, yeah?

Now, let's find $f(3)$. We know that $f(x) = 4 - x^2$. Substitute $x$ with $3$:

$f(3) = 4 - (3)^2$ $f(3) = 4 - 9$ $f(3) = -5$

Therefore, $f(3)$ equals -5. Be careful with the order of operations here! Remember to square the 3 before you subtract from 4. Many students make mistakes because of this simple point.

Now, we have our two key values: $g(3) = 18$ and $f(3) = -5$. Great! We’re halfway there. Now comes the final calculation.

Calculating (g-f)(3)

Now, to find $(g-f)(3)$, we subtract $f(3)$ from $g(3)$:

$(g-f)(3) = g(3) - f(3)$ $(g-f)(3) = 18 - (-5)$ $(g-f)(3) = 18 + 5$ $(g-f)(3) = 23$

So, $(g-f)(3) = 23$. This is our final answer! We've successfully navigated through the function operations. You can see how each step builds upon the previous one. This structured approach is what makes complex math problems manageable.

Matching the Answer to the Options

Now, let's look at the multiple-choice options provided in the prompt to find the one that gives us the same result. Remember, we need an expression that simplifies to 23. Let's analyze each option:

A. $6 - 3 - (4 + 3)^2$ B. $6 - 3 - (4 - 3^2)$ C. $6(3) - 4 + 3^2$ D. $6(3) - 4 - 3^2$

We know that the correct answer is 23, so we can calculate each of the options, step by step:

Option A: $6 - 3 - (4 + 3)^2$

First, solve the parentheses: $4 + 3 = 7$. Then square the result: $7^2 = 49$. Now, we have $6 - 3 - 49$. Simplifying further, $6 - 3 = 3$, and $3 - 49 = -46$. This does not match our answer, so Option A is incorrect.

Option B: $6 - 3 - (4 - 3^2)$

First, calculate the exponent: $3^2 = 9$. Then, solve the parentheses: $4 - 9 = -5$. Now, we have $6 - 3 - (-5)$. Then, $6 - 3 = 3$, and $3 - (-5) = 3 + 5 = 8$. This also does not match our result, so Option B is incorrect.

Option C: $6(3) - 4 + 3^2$

First, multiply: $6 * 3 = 18$. Then, calculate the exponent: $3^2 = 9$. Now we have $18 - 4 + 9$. Then, $18 - 4 = 14$, and $14 + 9 = 23$. This matches our answer, so Option C could be the correct one.

Option D: $6(3) - 4 - 3^2$

First, multiply: $6 * 3 = 18$. Then, calculate the exponent: $3^2 = 9$. Now we have $18 - 4 - 9$. Then, $18 - 4 = 14$, and $14 - 9 = 5$. This does not match our answer, so Option D is incorrect.

Therefore, we can confirm that Option C is the correct answer. This confirms the value of $(g-f)(3)$ we calculated.

Conclusion: Mastering Function Operations

So, there you have it, guys! We've successfully solved for $(g-f)(3)$. This problem is a classic example of how to work with functions and function notation. The key takeaways here are:

• Understand the notation: Know what $(g-f)(3)$ means. It's a specific calculation, not just a bunch of symbols.
• Break it down: Divide the problem into smaller, manageable steps (finding $g(3)$ and $f(3)$).
• Follow the order of operations: Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to avoid mistakes.
• Double-check your work: Always make sure your final answer makes sense in the context of the problem and verify it using the given options.

Mastering these concepts will greatly enhance your algebra skills. Keep practicing, and you'll become a pro at function operations in no time. If you got any questions, don’t hesitate to ask! Happy calculating!

This guide offers a detailed explanation for solving $(g-f)(3)$, making sure to provide clarity and understanding of the key concepts. We’ve covered everything from the basics to the final calculation, complete with the comparison with the available options, and we’ve also provided the necessary key takeaways. Remember, practice makes perfect, and with each problem you solve, you'll become more confident in your abilities. Keep up the great work, and happy learning!