Solving Composite Functions Step By Step F(g(x)) And G(f(x))

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Hey guys! Let's dive into the fascinating world of composite functions! Today, we're tackling a problem where we need to find the values of (f o g)(-3) and (g o f)(2), given the functions f(x) = 3 / (x + 2) (where x ≠ -2) and g(x) = 1 - 2x. We'll break it down step-by-step, so grab your pencils and let's get started!

Defining Composite Functions

Before we jump into the calculations, it's crucial to understand what composite functions are all about. Think of it like a machine where you feed in an input, and two functions work on it one after the other. The notation (f o g)(x) (read as "f of g of x") means we first apply the function g to x, and then we take the result and feed it into the function f. In other words, (f o g)(x) = f(g(x)).

Similarly, (g o f)(x) (read as "g of f of x") means we first apply the function f to x, and then we take the result and feed it into the function g. So, (g o f)(x) = g(f(x)). The order matters big time here, guys! Changing the order of the functions usually leads to a different final result. Understanding this order is key to mastering composite functions. We're essentially creating a chain reaction where the output of one function becomes the input of the next. This concept is super useful in various areas of mathematics and even in computer science, where you might have functions that process data in a specific sequence.

Now, why is this important? Composite functions allow us to build more complex operations from simpler ones. It's like having LEGO bricks – you can combine them in different ways to create amazing structures. In our case, f(x) and g(x) are our basic building blocks, and by composing them, we can explore new functional relationships. This gives us a powerful tool for modeling real-world situations where processes happen in stages. For example, imagine a discount applied to a sale price; that's a composition of two functions! So, let's keep this concept in mind as we move forward and tackle our specific problem.

A. Calculating (f o g)(-3)

Okay, let's find (f o g)(-3). Remember, this means we need to calculate f(g(-3)). We'll do this in two steps:

Step 1: Find g(-3)

First, we need to figure out what g(-3) is. We know that g(x) = 1 - 2x. So, we substitute x = -3 into the equation:

g(-3) = 1 - 2(-3) = 1 + 6 = 7

So, g(-3) = 7. This is the first piece of the puzzle! We've taken our initial input (-3) and processed it through the function g. Now, the output (7) becomes the input for the next step.

Step 2: Find f(g(-3)) which is f(7)

Now that we know g(-3) = 7, we can find f(g(-3)) which is the same as finding f(7). We're taking the result from the previous step and feeding it into the function f. We're given that f(x) = 3 / (x + 2). Let's substitute x = 7:

f(7) = 3 / (7 + 2) = 3 / 9 = 1/3

Therefore, f(7) = 1/3. We've now completed the second stage of our composite function journey. We took the output of g(-3) and used it as the input for f(x), resulting in a final value.

Conclusion for (f o g)(-3)

So, after these two steps, we can confidently say that (f o g)(-3) = f(g(-3)) = f(7) = 1/3. Awesome! We've successfully navigated our first composite function calculation. Remember, the key is to break it down into smaller, manageable steps. First, we work from the inside out, evaluating g(-3), and then we use that result as the input for f(x). This methodical approach will help you tackle even more complex composite function problems.

B. Calculating (g o f)(2)

Now, let's tackle (g o f)(2). This time, we need to calculate g(f(2)). Notice how the order has changed! This means we'll apply the functions in reverse order compared to part A. We'll start by finding f(2) and then use that result as the input for g(x).

Step 1: Find f(2)

We start by finding f(2). We know that f(x) = 3 / (x + 2). Substituting x = 2, we get:

f(2) = 3 / (2 + 2) = 3 / 4

So, f(2) = 3/4. We've successfully evaluated the inner function. This fraction will now be the input for our function g(x).

Step 2: Find g(f(2)) which is g(3/4)

Next, we need to find g(f(2)), which is the same as g(3/4). We know that g(x) = 1 - 2x. Let's substitute x = 3/4:

g(3/4) = 1 - 2(3/4) = 1 - 3/2 = 2/2 - 3/2 = -1/2

Therefore, g(3/4) = -1/2. We've now completed the entire process for this composite function. We took our initial input (2), processed it through f(x), and then took the result and fed it into g(x).

Conclusion for (g o f)(2)

Therefore, (g o f)(2) = g(f(2)) = g(3/4) = -1/2. Fantastic! We've successfully calculated (g o f)(2). Remember, the crucial difference between this and part A is the order of operations. We first applied f(x) and then g(x), highlighting the importance of paying close attention to the notation.

Key Takeaways and Why This Matters

Let's recap what we've learned, guys. We've successfully calculated both (f o g)(-3) and (g o f)(2). We've seen that:

  • (f o g)(-3) = 1/3
  • (g o f)(2) = -1/2

The most important thing to remember is the order of operations in composite functions. Always work from the inside out. This means evaluating the inner function first and then using its result as the input for the outer function. This concept is fundamental not just in math, but also in many other fields.

Why does this matter? Composite functions are more than just a mathematical exercise. They pop up in various real-world scenarios. For example:

  • Computer Programming: In programming, you often use functions that call other functions. This is essentially function composition. Think of it as building a complex program from smaller, reusable pieces.
  • Economics: Imagine a scenario where a tax is applied to the price of a product, and then a discount is applied to the taxed price. This is a composition of a tax function and a discount function.
  • Physics: In physics, you might have equations that describe the position of an object as a function of time, and then another equation that describes the time as a function of some other variable. Combining these equations involves function composition.

Practice Makes Perfect!

So, guys, the best way to truly understand composite functions is to practice, practice, practice! Try working through similar problems with different functions and different input values. Don't be afraid to make mistakes; that's how we learn! Break down each problem into smaller steps, and always remember to work from the inside out. You'll be a composite function pro in no time!

If you have any questions or want to explore more examples, feel free to ask! Keep practicing, and you'll master this concept in no time. You got this!