Solving Composite Functions A Step-by-Step Guide To (h∘f∘g)(x)

Hey guys! Today, we're diving into the fascinating world of function composition. It might sound intimidating, but trust me, it's a super useful concept in mathematics. We're going to break down a problem step-by-step to make sure you get a solid grasp of it. So, let's jump right in! We'll be tackling a problem that involves finding the composite function (h∘f∘g)(x) when we're given three different functions: f(x), g(x), and h(x). This is a classic example that really helps to illustrate how function composition works. Think of it like a mathematical assembly line – the output of one function becomes the input of the next! Understanding this process is key to mastering more advanced topics in calculus and other areas of mathematics.

The Problem: Decoding (h∘f∘g)(x)

So, here's the problem we're going to solve together:

If f(x) = 3x + 2, g(x) = 2x² - 4, and h(x) = (x + 3) / (2x - 1), where x ≠ 1/2, then what is (h∘f∘g)(x)?

This might look a bit daunting at first glance, but don't worry! We're going to break it down into manageable chunks. The symbol '∘' represents function composition, which essentially means we're plugging one function into another. In this case, (h∘f∘g)(x) means we first apply the function g to x, then we take that result and apply the function f to it, and finally, we take that result and apply the function h to it. It's like a chain reaction, each function building upon the previous one. The key to solving this kind of problem is to work from the inside out. We start with the innermost function, g(x), and then gradually work our way outwards, applying f(x) and then h(x) in sequence. This step-by-step approach helps to keep things organized and prevents us from getting lost in the complexity of the problem. We'll be using this strategy throughout the solution, so keep it in mind! Also, notice the condition x ≠ 1/2 for h(x). This is important because it tells us that the function h(x) is not defined when x = 1/2, as this would result in division by zero. We need to keep this in mind as we solve the problem, as it might affect the domain of the composite function.

Step 1: Finding (f∘g)(x)

The first step in finding (h∘f∘g)(x) is to determine (f∘g)(x). Remember, (f∘g)(x) means f(g(x)). This means we need to substitute the entire function g(x) into the function f(x) wherever we see an 'x'. It's like we're replacing the 'x' in f(x) with the entire expression for g(x). This might seem a little abstract, but it's a fundamental concept in function composition. Once you get the hang of it, it becomes second nature. Think of it as plugging a machine into another machine – the output of the first machine becomes the input of the second.

We know that f(x) = 3x + 2 and g(x) = 2x² - 4. So, to find f(g(x)), we replace the 'x' in f(x) with the expression for g(x), which is 2x² - 4. This gives us:

f(g(x)) = 3(2x² - 4) + 2

Now, we need to simplify this expression. We start by distributing the 3 across the parentheses:

f(g(x)) = 6x² - 12 + 2

Finally, we combine the constant terms:

f(g(x)) = 6x² - 10

So, we've found that (f∘g)(x) = 6x² - 10. This is a crucial step, as it forms the foundation for the next step in our calculation. We've essentially reduced the problem from a three-function composition to a two-function composition. Now, we'll use this result to find (h∘f∘g)(x). This process of breaking down a complex problem into smaller, more manageable steps is a common strategy in mathematics, and it's something you'll use again and again. By tackling each step individually, we can avoid getting overwhelmed and ensure that we arrive at the correct solution. So, let's move on to the next step and see how we can use this result to find the final answer!

Step 2: Finding (h∘f∘g)(x) which is h((f∘g)(x))

Now that we know (f∘g)(x) = 6x² - 10, we can find (h∘f∘g)(x), which is the same as h((f∘g)(x)). This means we need to substitute the expression we just found, 6x² - 10, into the function h(x) wherever we see an 'x'. Just like before, we're replacing the 'x' in h(x) with an entire expression, but this time, the expression is the result of our previous calculation. This highlights the chain-like nature of function composition – the output of one stage becomes the input of the next. This is where things might get a little trickier because h(x) is a fraction, but don't worry, we'll take it slow and steady. The key is to be meticulous and pay close attention to the details. We'll be substituting a slightly more complex expression into h(x), so it's important to keep track of everything and avoid making any careless mistakes. Remember, accuracy is just as important as understanding the process. So, let's dive in and see how it works!

We know that h(x) = (x + 3) / (2x - 1). So, to find h((f∘g)(x)), we replace the 'x' in h(x) with 6x² - 10. This gives us:

h((f∘g)(x)) = ((6x² - 10) + 3) / (2(6x² - 10) - 1)

Now, we need to simplify this expression. First, let's simplify the numerator and the denominator separately. In the numerator, we have:

(6x² - 10) + 3 = 6x² - 7

In the denominator, we first distribute the 2:

2(6x² - 10) - 1 = 12x² - 20 - 1

Then, we combine the constant terms:

12x² - 20 - 1 = 12x² - 21

So, we now have:

h((f∘g)(x)) = (6x² - 7) / (12x² - 21)

We're not quite done yet! We can simplify this fraction further by noticing that both the numerator and the denominator have a common factor of 3. We can factor out a 3 from both the numerator and the denominator:

h((f∘g)(x)) = (6x² - 7) / (3(4x² - 7))

Whoops! It looks like there was a slight error in the previous simplification. We cannot factor out a 3 from the numerator (6x² - 7). The denominator is correct: 12x² - 21 = 3(4x² - 7). So, let's correct our expression:

h((f∘g)(x)) = (6x² - 7) / (12x² - 21)

And that's our final answer! We've successfully found (h∘f∘g)(x). It was a bit of a journey, but we got there by breaking the problem down into smaller steps and carefully working through each step. This highlights the importance of patience and attention to detail in mathematics. Sometimes, the solution isn't immediately obvious, but by systematically applying the principles and techniques we've learned, we can arrive at the correct answer. Now, let's recap what we've done and make sure we've fully understood the process.

Final Answer: Putting It All Together

Therefore, (h∘f∘g)(x) = (6x² - 7) / (12x² - 21). Remember, we started with three functions, f(x), g(x), and h(x), and we wanted to find the composite function (h∘f∘g)(x). We achieved this by working from the inside out. First, we found (f∘g)(x) by substituting g(x) into f(x). Then, we took that result and substituted it into h(x) to find (h∘f∘g)(x). This process of function composition is a powerful tool in mathematics, allowing us to combine functions in interesting ways and create more complex models. It's used in a wide range of applications, from physics and engineering to computer science and economics. So, understanding function composition is not just about solving problems like this one – it's about building a foundation for more advanced mathematical concepts and real-world applications. I hope this explanation has been helpful and has made the concept of function composition a little clearer for you. Remember, the key to mastering mathematics is practice, so try working through some more examples on your own. The more you practice, the more confident you'll become in your ability to tackle these kinds of problems. And don't be afraid to ask for help if you get stuck – there are plenty of resources available, including your teachers, classmates, and online tutorials. Keep practicing, keep learning, and keep exploring the fascinating world of mathematics!

Key Takeaways and Practice

So, what are the key things to remember from this exercise? Firstly, function composition means applying one function to the result of another. We write (h∘f∘g)(x) to mean h(f(g(x))). Secondly, we work from the inside out. Evaluate g(x) first, then use that result as the input for f(x), and finally, use that result as the input for h(x). Thirdly, be careful with simplification. Make sure you distribute correctly and combine like terms accurately. And lastly, don't forget to consider any restrictions on the domain of the functions, like the x ≠ 1/2 condition in this problem. Understanding these key takeaways will help you approach future function composition problems with confidence. But remember, understanding is only half the battle – practice is essential for mastery. The more you practice, the more comfortable you'll become with the process, and the better you'll be able to apply it to different situations. So, to solidify your understanding, let's talk about some ways you can practice.

One great way to practice is to try working through similar problems with different functions. You can find plenty of examples in your textbook, online, or even create your own! Try changing the functions f(x), g(x), and h(x) and see how the composite function (h∘f∘g)(x) changes. This will help you develop a deeper understanding of how the different functions interact with each other. Another helpful exercise is to work backwards. Start with a composite function and try to break it down into its individual components. This can be a challenging but rewarding exercise that will really test your understanding of function composition. You can also try visualizing function composition graphically. Think about how the graphs of the individual functions are transformed when they are composed. This can provide a more intuitive understanding of the process and help you to remember the key concepts. And finally, don't be afraid to collaborate with others. Discuss the concepts with your classmates, work through problems together, and explain your reasoning to each other. This can be a great way to identify any gaps in your understanding and to learn from each other's insights. Remember, learning mathematics is a journey, and it's often more enjoyable and effective when you're working together with others. So, grab a friend, a textbook, and a pencil, and start practicing! The more you practice, the more confident and skilled you'll become in the world of function composition.