Hey there, math enthusiasts! Ever stumbled upon a problem that looks like a secret code? Today, we're diving into one of those – a function composition problem. Don't worry, it's not as intimidating as it sounds. We'll break it down step-by-step, making sure everyone can follow along. Our mission, should we choose to accept it, is to find the value of f(g(3)) given that f(x) = 2x + 3 and g(x) = 3x + 2. Ready to crack the code? Let's get started!
Unpacking the Functions: f(x) and g(x)
Before we can even think about f(g(3)), we need to understand what f(x) and g(x) are all about. Think of them as little machines. You feed a number into the machine (x), and it spits out a new number based on its special rule. f(x) = 2x + 3 is one such machine. This function machine takes your input (x), multiplies it by 2, and then adds 3. For example, if you feed in 1, the machine does (2 * 1) + 3, which equals 5. So, f(1) = 5. Simple enough, right?
Now, let's meet our second machine, g(x) = 3x + 2. This function machine has a slightly different rule. It takes your input (x), multiplies it by 3, and then adds 2. So, if we feed in 1 again, the machine calculates (3 * 1) + 2, which gives us 5. That means g(1) = 5. See how each machine has its own way of transforming numbers? Understanding these individual rules is key to tackling the bigger problem.
Understanding functions is crucial in mathematics, and these are linear functions, which means they produce a straight line when graphed. The beauty of functions lies in their ability to model real-world relationships. For instance, f(x) could represent the cost of buying x items if each item costs $2 and there's a $3 service fee. Similarly, g(x) could represent the total time it takes to complete x tasks if each task takes 3 minutes and there's a 2-minute setup time. By grasping the basics of functions, we unlock a powerful tool for solving problems and understanding the world around us. The heart of this problem lies in function composition, which we'll explore next.
The Art of Composition: What is f(g(x))?
Now, let's introduce a twist: what happens if we put these machines together? That's where the concept of function composition comes in. f(g(x)) looks a bit strange, but it's actually quite clever. It means we're feeding the output of the g(x) machine into the f(x) machine. It's like a two-step process. First, we take our input (x), run it through g(x), get an output, and then we take that output and feed it into f(x). Think of it like an assembly line where one machine's output becomes the next machine's input.
To get a clearer picture, let's break down how we'd find f(g(x)) in general. Remember, g(x) = 3x + 2. So, f(g(x)) really means f(3x + 2). Now, we need to take this entire expression, 3x + 2, and plug it in wherever we see an x in the f(x) function. Since f(x) = 2x + 3, we replace the x with (3x + 2), giving us f(g(x)) = 2(3x + 2) + 3. Now, we just need to simplify this expression. We distribute the 2, getting 6x + 4 + 3, and then combine like terms to get f(g(x)) = 6x + 7. This new function, 6x + 7, is the composition of f and g. It's a single machine that does the same thing as running a number through g and then f. Understanding this process of function composition opens up a world of possibilities for combining and manipulating functions.
Function composition isn't just a mathematical trick; it has practical applications in various fields. Imagine a scenario where a store offers a discount on an item, and then there's a sales tax added. If we let g(x) represent the price after the discount and f(x) represent the price after tax, then f(g(x)) would represent the final price you pay. This concept is also used in computer science, where functions are combined to create more complex programs. The ability to chain functions together allows for modularity and code reuse, making software development more efficient. So, mastering function composition is not only helpful for solving math problems but also for understanding real-world processes. Now, let's apply this knowledge to our specific problem and find f(g(3)).
Cracking the Code: Finding f(g(3))
Alright, we've laid the groundwork, and now it's time for the grand finale – finding the value of f(g(3)). Remember, we already know that f(x) = 2x + 3 and g(x) = 3x + 2. We also learned that f(g(x)) = 6x + 7. So, we have two ways to approach this. One method is to find g(3) first and then plug that result into f(x). The second method is to use the composite function we already found, f(g(x)) = 6x + 7, and simply plug in x = 3.
Let's start with the first method. First, we need to find g(3). We plug 3 into the g(x) function: g(3) = (3 * 3) + 2 = 9 + 2 = 11. So, g(3) = 11. Now, we take this result and plug it into the f(x) function: f(11) = (2 * 11) + 3 = 22 + 3 = 25. Therefore, f(g(3)) = 25. See how we worked from the inside out, first evaluating g(3) and then using that result in f(x)?
Now, let's try the second method, using the composite function f(g(x)) = 6x + 7. We simply plug in x = 3: f(g(3)) = (6 * 3) + 7 = 18 + 7 = 25. Guess what? We got the same answer! This demonstrates the power of function composition – it gives us a shortcut to solve the problem directly. Understanding different approaches to solving a problem is a valuable skill in mathematics. It not only reinforces your understanding but also provides a way to check your work. By using both methods, we've confidently cracked the code and found that f(g(3)) = 25.
Mission Accomplished: The Value of f(g(3)) Unveiled
We did it, guys! We successfully navigated the world of function composition and found the value of f(g(3)). We started by understanding the individual functions, f(x) and g(x), and then we explored the concept of function composition, f(g(x)). We even discovered a handy shortcut by finding the composite function directly. By using both methods, we confirmed that f(g(3)) = 25. This journey highlights the importance of breaking down complex problems into smaller, manageable steps. By tackling each piece individually, we can assemble the solution with confidence.
This exercise in mathematical problem-solving is not just about getting the right answer. It's about developing critical thinking skills, learning to approach problems systematically, and appreciating the elegance of mathematical concepts. The power of functions lies in their ability to model relationships and transformations, providing a framework for understanding various phenomena. From calculating discounts to modeling physical processes, functions are a fundamental tool in mathematics and beyond. So, the next time you encounter a function composition problem, remember the steps we took today: understand the individual functions, grasp the concept of composition, and break down the problem step-by-step. With practice and perseverance, you'll become a master code-cracker in no time!