Inverse Function & Sum Of Functions: Solved!

by ADMIN 45 views

Hey guys! Let's break down these math problems step by step. We're going to tackle finding the inverse of a function and then dive into adding two functions together. Get ready to sharpen your pencils and your minds!

Finding the Inverse of a Function: g(x) = 28 - x

Okay, so the first part of our mission is to find the inverse of the function g(x) = 28 - x. You might be asking, "What even is an inverse function?" Well, simply put, the inverse function "undoes" what the original function does. Think of it like this: if you put a number into the original function and get a result, the inverse function will take that result and give you back your original number.

Now, let's get into the nitty-gritty of how to find this inverse. There's a pretty standard method we can follow, and I'll walk you through it. First things first, we're going to replace g(x) with y. This just makes the equation a little easier to work with. So, we have:

y = 28 - x

Next up, and this is the key step, we're going to swap x and y. Yep, you read that right! We're switching their places. This is the heart of finding the inverse because it reflects the "undoing" action we talked about earlier. Our equation now looks like this:

x = 28 - y

Now, our goal is to get y by itself on one side of the equation. This is just basic algebra, so we can do it! We need to isolate y. Let's subtract 28 from both sides:

x - 28 = -y

Almost there! We've got -y, but we want just y. So, we multiply both sides by -1 (or you can think of it as dividing both sides by -1, same thing):

-x + 28 = y

Or, if we rearrange it a bit to look nicer:

y = 28 - x

So, what we've found here is the inverse function. But to be super official, we use a special notation for inverse functions. We write it as g⁻¹(x). That little -1 up there looks like an exponent, but it's not! It just means "inverse." So, we can write our final answer as:

g⁻¹(x) = 28 - x

And that's it! We've found the inverse function. Looking back at the original multiple-choice options, we see that b. 28 - x is indeed the correct answer. High five!

To make sure we really understand this, let's recap the steps we took:

  1. Replace g(x) with y.
  2. Swap x and y.
  3. Solve for y.
  4. Replace y with g⁻¹(x). (This is just the final notation step).

Keep these steps in mind, and you'll be finding inverse functions like a pro in no time!

Adding Functions: Finding (f + g)(x) when f(x) = 4x² + 2x - 2 and g(x) = 3x² - 4x + 1

Alright, let's shift gears and tackle the second part of our math adventure: adding functions. This might sound intimidating, but trust me, it's actually pretty straightforward. We're given two functions:

f(x) = 4x² + 2x - 2

g(x) = 3x² - 4x + 1

And we need to find (f + g)(x). What does this even mean? Well, (f + g)(x) simply means that we're going to add the function f(x) to the function g(x). That's it! No tricks, no hidden agendas. We're just adding them together.

So, let's do it! We'll write out the two functions and put a plus sign in between them:

(f + g)(x) = (4x² + 2x - 2) + (3x² - 4x + 1)

Now, we need to combine like terms. Remember those from algebra? Like terms are terms that have the same variable raised to the same power. So, we'll group together the x² terms, the x terms, and the constant terms (the ones without any x's).

Let's start with the x² terms: we have 4x² from f(x) and 3x² from g(x). Adding them together, we get:

4x² + 3x² = 7x²

Next, let's look at the x terms: we have 2x from f(x) and -4x from g(x). Adding these gives us:

2x - 4x = -2x

Finally, we have the constant terms: -2 from f(x) and +1 from g(x). Adding these together, we get:

-2 + 1 = -1

Now, we just put all these pieces together to get our final answer:

(f + g)(x) = 7x² - 2x - 1

And there you have it! We've successfully added the two functions together. It's like combining ingredients in a recipe – you just put them all in the bowl and mix them up!

To recap the steps for adding functions:

  1. Write out the expression (f + g)(x) = f(x) + g(x). This helps keep things organized.
  2. Substitute the actual expressions for f(x) and g(x). This is where the functions come into play.
  3. Combine like terms. This is the crucial step where you simplify the expression.

Keep practicing, and adding functions will become second nature to you!

Wrapping Up

So, we've conquered two important math concepts today: finding the inverse of a function and adding functions together. We saw how finding the inverse involves swapping variables and solving for y, and how adding functions is all about combining like terms. These are fundamental skills in algebra and calculus, so mastering them is super important for your mathematical journey. Remember, math isn't about memorizing formulas; it's about understanding the process and the why behind it. Keep practicing, keep asking questions, and you'll be amazed at what you can achieve! You got this, guys!