Solving Composite Functions G(f(x)) And G(f(2)) Explained

Hey guys! Ever found yourself staring at function compositions and feeling a bit lost? No worries, we've all been there! Let's break down a super common type of problem in math: composite functions. We're going to tackle a specific example step-by-step, making sure you understand exactly what's going on. We'll be looking at how to find (g°f)(x) and then how to evaluate (g°f)(2). So, buckle up and let's dive in!

Understanding Composite Functions

Before we jump into the problem itself, it's crucial to grasp the fundamental idea of composite functions. Think of it like a machine where you feed in an input, and then another machine processes the output of the first one. In mathematical terms, if you have two functions, f(x) and g(x), the composite function (g°f)(x) means you're first applying the function f to x, and then you're taking the result of that and feeding it into the function g. This "chain reaction" is what makes composite functions so interesting and powerful. To really nail this down, let's visualize it: Imagine f(x) is a blender that turns ingredients into a smoothie, and g(x) is an oven that bakes things. If you want to make smoothie muffins, you'd first blend your ingredients using f(x), and then bake the smoothie batter using g(x). The order matters here! Baking the ingredients first and then trying to blend them wouldn't give you the same result. The same principle applies to mathematical functions. The order of composition is key to getting the correct answer. We express the composition as (g°f)(x), which is read as "g of f of x." It's a shorthand way of saying "apply f to x, and then apply g to the result." Another way to write this is g(f(x)), which visually emphasizes that f(x) is the input for the function g. Understanding this notation is crucial because it tells you the order in which the functions should be applied. Ignoring this order is a common mistake, so always double-check which function is being applied first. Now that we have a solid grasp of what composite functions are, we're ready to tackle our specific problem. Remember, the key is to break it down step-by-step and understand what each function is doing at each stage. With practice, these types of problems become much easier to solve!

Problem Setup: f(x) and g(x)

Alright, let's get down to business! Our problem gives us two functions: f(x) = 2x + 3 and g(x) = 3x² - 5x + 1. These are the stars of our show today. Function f(x) is a linear function – it takes an input (x), multiplies it by 2, and then adds 3. It's a pretty straightforward operation. Function g(x), on the other hand, is a quadratic function. It takes an input (x), squares it, multiplies the squared value by 3, subtracts 5 times the original input, and then adds 1. Quadratic functions have a bit more going on, but don't worry, we'll handle it step-by-step. The problem asks us to find two things: First, we need to find the expression for the composite function (g°f)(x). This means we need to figure out what happens when we apply f(x) first and then apply g(x) to the result. In other words, we're substituting the entire function f(x) into the 'x' of the function g(x). This might sound a little abstract right now, but we'll make it crystal clear in the next section. Second, after we find the expression for (g°f)(x), we need to evaluate it at x = 2. This means we'll plug in the value 2 for 'x' in our (g°f)(x) expression and simplify to get a numerical answer. This step is all about applying our understanding of function evaluation. Before we dive into the calculations, let's take a moment to appreciate the beauty of these functions. Functions are the building blocks of mathematics, and they allow us to model and understand the world around us. By mastering function composition, you're not just learning a mathematical technique; you're building a powerful tool for problem-solving in various fields. Now, with our functions defined and our goals clear, let's get to the fun part: the calculations!

Finding (g°f)(x): Step-by-Step

Okay, guys, this is where the magic happens! We're going to find the expression for (g°f)(x). Remember, this means we're plugging f(x) into g(x). Think of it like this: wherever you see an 'x' in g(x), you're going to replace it with the entire expression for f(x), which is 2x + 3. Let's write down g(x) again to have it handy: g(x) = 3x² - 5x + 1. Now, we're going to substitute f(x) = 2x + 3 into g(x). This gives us: g(f(x)) = 3(2x + 3)² - 5(2x + 3) + 1. See how we've replaced every 'x' in g(x) with (2x + 3)? This is the key step in finding the composite function. Now, we need to simplify this expression. The first thing we need to do is expand (2x + 3)². Remember that (a + b)² = a² + 2ab + b². Applying this to our expression, we get: (2x + 3)² = (2x)² + 2(2x)(3) + 3² = 4x² + 12x + 9. So now our expression looks like this: g(f(x)) = 3(4x² + 12x + 9) - 5(2x + 3) + 1. Next, we distribute the 3 and the -5: g(f(x)) = 12x² + 36x + 27 - 10x - 15 + 1. Finally, we combine like terms: g(f(x)) = 12x² + (36x - 10x) + (27 - 15 + 1) = 12x² + 26x + 13. Woohoo! We've done it! We've found the expression for (g°f)(x): (g°f)(x) = 12x² + 26x + 13. This is a new quadratic function that represents the composition of f(x) and g(x). Take a moment to appreciate the journey we've taken. We started with two individual functions, and through the magic of composition, we've created a new one. Now that we have this expression, we're ready for the final step: evaluating it at x = 2. Get ready to plug in and chug!

Evaluating (g°f)(2)

Alright, team, we're in the home stretch! We've already found that (g°f)(x) = 12x² + 26x + 13. Now, we need to find (g°f)(2). This is a straightforward substitution problem. All we need to do is replace every 'x' in our expression for (g°f)(x) with the number 2. So, we have: (g°f)(2) = 12(2)² + 26(2) + 13. Now, let's simplify this step-by-step. First, we calculate 2²: 2² = 4. So our expression becomes: (g°f)(2) = 12(4) + 26(2) + 13. Next, we perform the multiplications: 12(4) = 48 and 26(2) = 52. Now we have: (g°f)(2) = 48 + 52 + 13. Finally, we add the numbers together: 48 + 52 + 13 = 113. Boom! We've got our answer! (g°f)(2) = 113. This means that if we first apply the function f to the number 2, and then apply the function g to the result, we get the number 113. Isn't that neat? We've taken the number 2, put it through our function composition machine, and out popped 113. This is the power of function composition in action. Now, let's take a moment to reflect on what we've accomplished. We started with a problem involving two functions, f(x) and g(x). We learned how to find the composite function (g°f)(x) by substituting f(x) into g(x). We simplified the resulting expression and obtained a new function. Then, we evaluated this new function at x = 2 to get a numerical answer. We've covered a lot of ground, and you've done a fantastic job following along. Remember, practice makes perfect, so try working through similar problems to solidify your understanding. You've now got a solid foundation in composite functions, and you're well on your way to mastering more advanced mathematical concepts!

Key Takeaways and Final Thoughts

Alright, let's recap what we've learned today and leave you with some key takeaways. We've successfully navigated the world of composite functions, specifically focusing on finding (g°f)(x) and evaluating (g°f)(2). The most crucial thing to remember is the order of operations in function composition. (g°f)(x) means applying f(x) first and then applying g(x) to the result. This is a fundamental concept that underpins the entire process. We also saw how to substitute one function into another. This involves replacing the 'x' in the outer function (in our case, g(x)) with the entire expression for the inner function (f(x)). This substitution is the heart of finding the composite function. After the substitution, simplification is key. This often involves expanding squared terms, distributing, and combining like terms. Don't rush this step – accuracy is crucial. A small mistake in simplification can lead to a completely wrong answer. Finally, we learned how to evaluate a composite function at a specific value. This is a straightforward substitution process where you replace 'x' with the given value and simplify. Remember, composite functions are not just abstract mathematical concepts. They have real-world applications in various fields, including computer science, engineering, and physics. They allow us to model complex systems by breaking them down into smaller, more manageable steps. So, by mastering composite functions, you're not just learning a math skill; you're gaining a powerful tool for problem-solving in a variety of contexts. Keep practicing, keep exploring, and keep pushing your mathematical boundaries. You've got this! And hey, if you ever get stuck, just remember the smoothie muffin analogy – blend first, then bake! That should help you keep the order of operations straight. Until next time, happy calculating!