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

by Andrew McMorgan 39 views

Hey Plastik Magazine readers! Let's dive into a math problem today that might seem a bit tricky at first, but we'll break it down together. We're tackling a question that involves function operations, specifically finding an equivalent expression for (g-f)(3) when given two functions: f(x) = 4 - x^2 and g(x) = 6x. Don't worry if that looks like a jumble of letters and numbers – we'll make sense of it all. So grab your favorite beverage, maybe a notepad and pen, and let's get started!

Understanding Function Operations

Before we jump into the specific problem, let's quickly review what function operations mean. Basically, just like we can add, subtract, multiply, and divide numbers, we can also do these operations with functions. When we see (g - f)(x), it means we're subtracting the function f(x) from the function g(x). In other words, (g - f)(x) = g(x) - f(x). This is a crucial concept to grasp because it forms the foundation for solving our problem. We're not just dealing with individual numbers here; we're working with entire expressions that define how the functions behave. Think of it like this: each function is a little machine that takes an input (x) and spits out an output based on its own rule. When we subtract one function from another, we're essentially comparing and contrasting these machines and their outputs. So, with this understanding of function operations in our mental toolkit, we're ready to tackle the specifics of our problem and find out what (g-f)(3) really means.

Breaking Down the Problem: Finding (g-f)(3)

Okay, now let's get to the heart of the matter. We need to find the equivalent expression for (g - f)(3). Remember, this means we first need to find (g - f)(x) and then substitute x = 3 into that expression. So, the first step is to figure out what g(x) - f(x) actually looks like. We know that g(x) = 6x and f(x) = 4 - x^2. Therefore, (g - f)(x) = 6x - (4 - x^2). Now, be careful with that minus sign! We need to distribute it to both terms inside the parentheses. This gives us (g - f)(x) = 6x - 4 + x^2. See how the -x^2 became +x^2 because of the subtraction? This is a common place to make a mistake, so always double-check those signs. Now we have a simplified expression for (g - f)(x), which is x^2 + 6x - 4. The next step is where the magic happens: we substitute x = 3 into this expression. This means we replace every 'x' with a '3'. This will give us a numerical value, which represents the value of the combined function (g - f) when the input is 3. So, let's move on to the substitution and see what we get!

Substituting x = 3

Alright, we've found that (g - f)(x) = x^2 + 6x - 4. Now comes the moment of truth: let's substitute x = 3 into this expression. This means we replace every instance of 'x' with the number '3'. So, we get (g - f)(3) = (3)^2 + 6(3) - 4. Notice how we've carefully put parentheses around the 3 to make it clear that we're squaring 3 and multiplying 6 by 3. Now we just need to follow the order of operations (PEMDAS/BODMAS) to simplify this expression. First, we handle the exponent: 3^2 is 3 * 3, which equals 9. Next, we do the multiplication: 6 * 3 equals 18. So now we have (g - f)(3) = 9 + 18 - 4. Finally, we do the addition and subtraction from left to right: 9 + 18 = 27, and then 27 - 4 = 23. Therefore, (g - f)(3) = 23. This is our final numerical answer! But remember, the question asked for an equivalent expression, not just the numerical value. So we need to look at the answer choices and see which one simplifies to 23. Let's move on to analyzing those options and finding the correct match.

Analyzing the Answer Choices

Okay, we've determined that (g - f)(3) = 23. Now, let's look at those answer choices and see which one gives us the same result. This is where careful calculation and attention to detail are key. We'll go through each option step-by-step, simplifying them to see if they equal 23.

  • Option A: 6 - 3 - (4 + 3)^2 First, we simplify inside the parentheses: (4 + 3) = 7. Then we square it: 7^2 = 49. So the expression becomes 6 - 3 - 49. Now we subtract from left to right: 6 - 3 = 3, and then 3 - 49 = -46. This does not equal 23, so option A is incorrect.
  • Option B: 6 - 3 - (4 - 3^2) First, we handle the exponent inside the parentheses: 3^2 = 9. So the expression inside the parentheses becomes (4 - 9) = -5. Now we have 6 - 3 - (-5). Remember that subtracting a negative is the same as adding, so this becomes 6 - 3 + 5. Subtracting from left to right: 6 - 3 = 3, and then 3 + 5 = 8. This does not equal 23, so option B is also incorrect.
  • Option C: 6(3) - 4 + 3^2 First, we do the multiplication: 6(3) = 18. Then we handle the exponent: 3^2 = 9. So the expression becomes 18 - 4 + 9. Now we add and subtract from left to right: 18 - 4 = 14, and then 14 + 9 = 23. Bingo! This option equals 23, so it looks like we've found our answer.
  • Option D: 6(3) - 4 - 3^2 Let's just double-check this one to be sure. First, we do the multiplication: 6(3) = 18. Then we handle the exponent: 3^2 = 9. So the expression becomes 18 - 4 - 9. Subtracting from left to right: 18 - 4 = 14, and then 14 - 9 = 5. This does not equal 23, so option D is incorrect.

Therefore, the equivalent expression for (g - f)(3) is option C: 6(3) - 4 + 3^2.

Key Takeaways and Tips

Great job, guys! We successfully navigated this function operation problem. Let's recap the key steps and some helpful tips:

  • Understand Function Operations: Make sure you know what (g - f)(x), (g + f)(x), (g * f)(x), and (g / f)(x) mean. It's all about performing operations on the function expressions themselves.
  • Distribute Negative Signs Carefully: When subtracting functions, remember to distribute the negative sign to all terms in the function being subtracted. This is a common source of errors.
  • Follow the Order of Operations (PEMDAS/BODMAS): Exponents, multiplication, division, addition, subtraction – follow the correct order to avoid mistakes.
  • Substitute Carefully: When substituting a value for x, be sure to replace every instance of x in the expression.
  • Check Your Work: If you have time, quickly recalculate to make sure you haven't made any arithmetic errors. Especially on multiple-choice questions, plugging your answer back into the original equation can confirm its validity.
  • Practice Makes Perfect: The more you solve these types of problems, the more comfortable you will become with functions and how to manipulate them. So don't give up, practice, practice, practice!

I hope this breakdown was helpful! Remember, math can be challenging, but by breaking it down into smaller steps and understanding the underlying concepts, you can tackle any problem. Keep practicing, and you'll become a function operation master in no time! Stay tuned for more math adventures here at Plastik Magazine!