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

Hey there, math enthusiasts! Today, we're diving into a cool algebra problem. Our mission? To find the equivalent expression for (g - f)(3). This means we need to figure out the value of the expression when x is equal to 3, given two functions: f(x) = 4 - x^2 and g(x) = 6x. Don't worry, it's easier than it sounds, and we'll break it down step by step.

Understanding the Problem

First things first, let's make sure we're all on the same page. We're dealing with function subtraction, which means we're subtracting the value of the function f(x) from the value of the function g(x). When we see (g - f)(3), it's the same as saying g(3) - f(3). In other words, we need to find the value of g(x) when x = 3, find the value of f(x) when x = 3, and then subtract the latter from the former. Simple, right?

We'll start by finding the values of f(3) and g(3) separately before calculating (g - f)(3). This is a straightforward process of substitution and evaluation. Remember, the key is to replace x with the given value (in this case, 3) in each function and then simplify. Let's get our hands dirty with the calculations!

Calculating f(3)

Okay, let's find out what f(3) equals. We know that f(x) = 4 - x^2. Now, we just need to substitute x with 3: f(3) = 4 - (3)^2. Following the order of operations (PEMDAS/BODMAS), we first square 3, which gives us 9. Then, we subtract 9 from 4. So, f(3) = 4 - 9 = -5. Easy peasy, right? We now have the value of f(3). Keep that in mind – it's an important piece of our puzzle!

This step is all about applying the function's rule to a specific input. The function f(x) tells us to subtract the square of the input from 4. By doing this, we're mapping the input value of 3 to an output value of -5. It's a fundamental concept in understanding how functions work. Remember that each function has its unique rule that dictates how inputs are transformed into outputs. Here, our rule involves squaring and subtracting, leading us to the value of -5 when the input is 3. With this, we're one step closer to cracking the complete expression (g - f)(3)!

Calculating g(3)

Alright, time to find g(3). We're given that g(x) = 6x. To find g(3), we replace x with 3: g(3) = 6 * 3. Multiplying 6 by 3 gives us 18. Therefore, g(3) = 18. We've now calculated both f(3) and g(3). The hard part is over; we just need to put it all together to find our final answer.

Here, we're using the function g(x), which has a rule of multiplying the input by 6. Substituting 3 into this function means we're applying this multiplication rule to the number 3, giving us 18. This shows how different functions transform the same input in different ways. The value of g(3) is essential for our final calculation of (g - f)(3). With g(3) = 18 and f(3) = -5, we are ready to find the difference between the two values, leading us to the answer. This illustrates the power of function notation in representing mathematical relationships concisely.

Finding (g - f)(3)

Now, the grand finale! We've got f(3) = -5 and g(3) = 18. Recall that (g - f)(3) = g(3) - f(3). So, we substitute the values we found: (g - f)(3) = 18 - (-5). Remember that subtracting a negative number is the same as adding its positive counterpart. Therefore, 18 - (-5) = 18 + 5 = 23. And there you have it! (g - f)(3) = 23.

This is where all the pieces of the puzzle come together. By understanding function subtraction, we were able to determine the value of the combined expression. The result, 23, is the difference between the outputs of functions g(x) and f(x) when the input is 3. This process demonstrates the importance of understanding function notation and order of operations. Each step, from finding f(3) and g(3) to the final subtraction, highlights the practical application of algebraic concepts. Successfully solving this problem underscores the power of breaking down complex expressions into manageable steps.

Matching with the Options

Now that we know the answer is 23, let's look at the options provided to identify the correct one. We need to find an expression that, when evaluated, gives us 23. We know that (g - f)(3) = g(3) - f(3) = 6(3) - (4 - 3^2). Let's evaluate the options:

A. 6 - 3 - (4 + 3)^2 = 3 - 49 = -46 B. 6 - 3 - (4 - 3^2) = 3 - (4 - 9) = 3 - (-5) = 8 C. 6(3) - 4 + 3^2 = 18 - 4 + 9 = 23 D. 6(3) - 4 - 3^2 = 18 - 4 - 9 = 5

Option C matches our result. So, the correct expression is C.

Conclusion

So, guys, we've successfully found the equivalent expression for (g - f)(3). It was a fun journey, wasn't it? We first understood what the problem was asking, then we found the individual values of f(3) and g(3), and finally, we subtracted f(3) from g(3) to get our answer. The correct expression from the options is 6(3) - 4 + 3^2. Keep practicing, and you'll ace these problems in no time! Remember, the key is to break down the problem into smaller, easier steps. Keep up the great work, and keep exploring the wonderful world of math! Remember, understanding functions and how to manipulate them is crucial for more complex problems. Great job, everyone, and happy solving!