Function Operations: Solve (f-g)(1), (f+g)(2), (fg)(3), (f/g)(4)

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Hey guys! Today, we're diving into the world of function operations. Function operations might sound intimidating, but trust me, it's just a fancy way of saying we're going to add, subtract, multiply, and divide functions. We'll tackle a pretty classic problem: Given two functions, f(x) and g(x), we need to find the values of combined functions like (f - g)(1), (f + g)(2), (fg)(3), and (f/g)(4). So, let's break it down step by step and make sure we understand exactly what's going on. These problems are super common in algebra and calculus, so nailing this now will definitely pay off later. Let's jump right in and make math fun!

Understanding Function Operations

Before we jump into solving the specific problem, let's make sure we're all on the same page about what function operations actually mean. At their core, function operations are ways to combine two or more functions to create a new function. Think of it like this: you've got your individual ingredients (the functions), and you're going to mix them together in different ways to get new dishes (the combined functions).

Here’s a quick rundown of the basic operations:

  • Addition: (f + g)(x) = f(x) + g(x). We simply add the two functions together.
  • Subtraction: (f - g)(x) = f(x) - g(x). We subtract the second function from the first.
  • Multiplication: (fg)(x) = f(x) * g(x). We multiply the two functions.
  • Division: (f/g)(x) = f(x) / g(x), but there's a catch! We need to make sure that g(x) isn't equal to zero, because we can't divide by zero. This gives rise to a domain restriction, which we'll need to consider.

The key thing to remember is that these operations create new functions. So, for example, if you add two functions together, you get a brand-new function that behaves differently from the originals. Now, with the basics down, let's get into the nitty-gritty of how to actually solve these problems.

Problem Setup

Okay, let's get our hands dirty with the actual problem. We're given two functions:

  • f(x) = 8x / (x - 2)
  • g(x) = 2x + 3

And we need to find the values of the following:

  • (f - g)(1)
  • (f + g)(2)
  • (fg)(3)
  • (f/g)(4)

So, what's the plan of attack? First, we'll find the combined functions (f - g)(x), (f + g)(x), (fg)(x), and (f/g)(x). This will give us general formulas. Then, we'll simply plug in the given x values (1, 2, 3, and 4) into these combined functions to get our answers. Seems straightforward, right? Let's start with the first operation: subtraction.

Finding (f - g)(1)

To find (f - g)(1), we first need to determine (f - g)(x). Remember, this means we're subtracting g(x) from f(x). So we have:

(f - g)(x) = f(x) - g(x)

Now, let's plug in the actual functions:

(f - g)(x) = [8x / (x - 2)] - [2x + 3]

This looks a little messy, but don't worry! We can clean it up. To subtract these, we need a common denominator. The denominator of the first term is (x - 2), and we can think of the second term as having a denominator of 1. So, we need to multiply the second term by (x - 2) / (x - 2):

(f - g)(x) = [8x / (x - 2)] - [(2x + 3) * (x - 2) / (x - 2)]

Now we can combine them:

(f - g)(x) = [8x - (2x + 3)(x - 2)] / (x - 2)

Let's expand the numerator:

(f - g)(x) = [8x - (2x² - 4x + 3x - 6)] / (x - 2)

(f - g)(x) = [8x - (2x² - x - 6)] / (x - 2)

Distribute the negative sign:

(f - g)(x) = (8x - 2x² + x + 6) / (x - 2)

Combine like terms:

(f - g)(x) = (-2x² + 9x + 6) / (x - 2)

Alright, we've got (f - g)(x)! Now we can find (f - g)(1) by plugging in x = 1:

(f - g)(1) = [-2(1)² + 9(1) + 6] / (1 - 2)

(f - g)(1) = (-2 + 9 + 6) / (-1)

(f - g)(1) = 13 / (-1)

(f - g)(1) = -13

So, (f - g)(1) = -13. Awesome! Let's move on to the next operation.

Finding (f + g)(2)

Next up, we need to find (f + g)(2). This time, we're adding the functions. So, first, let's find (f + g)(x):

(f + g)(x) = f(x) + g(x)

Plug in the functions:

(f + g)(x) = [8x / (x - 2)] + [2x + 3]

Again, we need a common denominator, which is (x - 2). So, we multiply the second term by (x - 2) / (x - 2):

(f + g)(x) = [8x / (x - 2)] + [(2x + 3)(x - 2) / (x - 2)]

Combine the fractions:

(f + g)(x) = [8x + (2x + 3)(x - 2)] / (x - 2)

Expand the numerator:

(f + g)(x) = [8x + (2x² - 4x + 3x - 6)] / (x - 2)

(f + g)(x) = [8x + 2x² - x - 6] / (x - 2)

Combine like terms:

(f + g)(x) = (2x² + 7x - 6) / (x - 2)

Now we have (f + g)(x). To find (f + g)(2), we plug in x = 2:

(f + g)(2) = [2(2)² + 7(2) - 6] / (2 - 2)

(f + g)(2) = (8 + 14 - 6) / 0

Uh oh! We have a division by zero. This means that (f + g)(2) is undefined. Remember, division by zero is a big no-no in math.

Finding (fg)(3)

Now, let's tackle multiplication. We want to find (fg)(3), so we first need (fg)(x). This means we're multiplying f(x) and g(x):

(fg)(x) = f(x) * g(x)

Plug in the functions:

(fg)(x) = [8x / (x - 2)] * [2x + 3]

This is a bit simpler since we're just multiplying. We can rewrite this as:

(fg)(x) = [8x(2x + 3)] / (x - 2)

Expand the numerator:

(fg)(x) = (16x² + 24x) / (x - 2)

Now we have (fg)(x). Let's find (fg)(3) by plugging in x = 3:

(fg)(3) = [16(3)² + 24(3)] / (3 - 2)

(fg)(3) = (16 * 9 + 72) / 1

(fg)(3) = (144 + 72) / 1

(fg)(3) = 216

So, (fg)(3) = 216. We're on a roll!

Finding (f/g)(4)

Last but not least, we need to find (f/g)(4). This involves division, so we need to be careful about the domain. First, let's find (f/g)(x):

(f/g)(x) = f(x) / g(x)

Plug in the functions:

(f/g)(x) = [8x / (x - 2)] / [2x + 3]

To divide fractions, we multiply by the reciprocal of the second fraction:

(f/g)(x) = [8x / (x - 2)] * [1 / (2x + 3)]

(f/g)(x) = 8x / [(x - 2)(2x + 3)]

Now we have (f/g)(x). Let's plug in x = 4 to find (f/g)(4):

(f/g)(4) = 8(4) / [(4 - 2)(2(4) + 3)]

(f/g)(4) = 32 / [2(8 + 3)]

(f/g)(4) = 32 / (2 * 11)

(f/g)(4) = 32 / 22

We can simplify this fraction by dividing both the numerator and denominator by 2:

(f/g)(4) = 16 / 11

So, (f/g)(4) = 16/11. We did it!

Conclusion

Alright guys, we've successfully found all the values! To recap:

  • (f - g)(1) = -13
  • (f + g)(2) is undefined
  • (fg)(3) = 216
  • (f/g)(4) = 16/11

We walked through adding, subtracting, multiplying, and dividing functions, and even handled a tricky division by zero situation. The key takeaways here are understanding what each operation means, finding a common denominator when adding or subtracting, and always being mindful of domain restrictions, especially when dividing. Mastering these function operations opens the door to more complex math concepts, so great job sticking with it! You've tackled a significant challenge and come out on top. Keep practicing, and you'll become a function operation pro in no time! Remember, math can be fun when we break it down step by step, just like we did today. Keep exploring, keep learning, and most importantly, keep enjoying the process! 🚀