Evaluate (f ∘ H)(x) At X = -1: A Step-by-Step Solution

Let's dive into evaluating the composite function (f ∘ h)(x) when x = -1, given the functions f(x) = x³ + 9x, g(x) = √(2x), and h(x) = 8x + 5. This might sound intimidating, but we'll break it down step by step to make it super clear and easy to follow. We will explore the concept of composite functions, and how to solve them with a practical example. So, let’s get started and make math fun!

Understanding Composite Functions

First off, let's quickly refresh what a composite function actually is. Think of it like this: it's a function within a function. When we see (f ∘ h)(x), it means we're plugging the function h(x) into the function f(x). It’s like a mathematical assembly line where the output of one function becomes the input of another. The notation (f ∘ h)(x) is read as "f of h of x," emphasizing this sequential application. To evaluate a composite function, we must first find the value of the inner function, h(x) in this case, and then substitute that value into the outer function, f(x). This process highlights the interconnectedness of functions, where the result of one directly influences the next, creating a chain of operations. Understanding the order of operations is crucial in composite functions, as reversing the order can lead to completely different results. Therefore, always start with the innermost function and work your way outwards to ensure accurate evaluation.

Step 1: Find h(x)

Our first task is to find h(x) when x = -1. Remember, h(x) = 8x + 5. This is a straightforward linear function, which makes it easy to evaluate. We substitute -1 for x in the expression for h(x) and simplify. Let's do it together: h(-1) = 8(-1) + 5. Multiplying 8 by -1 gives us -8, so now we have h(-1) = -8 + 5. Adding -8 and 5, we get -3. So, h(-1) = -3. This value is crucial because it becomes the input for our next step, where we'll use it in the function f(x). Understanding how to correctly evaluate h(x) is fundamental because it lays the groundwork for the rest of the problem. It's like setting up the first domino in a chain reaction; if you get it wrong, the rest of the sequence will be off. Accuracy in this step is paramount to achieving the correct final answer. Make sure to double-check your arithmetic to avoid simple mistakes that can derail the entire process.

Step 2: Plug h(-1) into f(x)

Now that we know h(-1) = -3, we can move on to the second part of our journey: plugging this value into f(x). Remember, f(x) = x³ + 9x. This means we're going to replace x in f(x) with -3. So, we have f(h(-1)) = f(-3) = (-3)³ + 9(-3). This step involves both exponentiation and multiplication, so we need to follow the order of operations carefully. First, let's calculate (-3)³. This means -3 multiplied by itself three times: -3 * -3 * -3. -3 * -3 equals 9, and then 9 * -3 equals -27. So, (-3)³ = -27. Next, we calculate 9(-3), which is -27. Now we can rewrite our equation as f(-3) = -27 + (-27). Adding these two negative numbers gives us f(-3) = -54. This result is the value of the composite function (f ∘ h)(x) when x = -1. This step illustrates the core of composite functions: taking the result from one function and feeding it into another. The power of composite functions lies in their ability to create complex relationships by chaining simpler functions together. It's like building a machine where each part performs a specific task, and the overall output is the result of all the parts working in sequence.

Step 3: The Final Answer

We've done the heavy lifting, and now we're at the final step! We found that f(h(-1)) = -54. That's it! The value of the composite function (f ∘ h)(x) when x = -1 is -54. This concise answer is the culmination of all our efforts, from understanding composite functions to carefully evaluating each step. It's a great feeling to arrive at a solution, and it's a testament to the power of breaking down complex problems into manageable parts. Remember, mathematics is like a puzzle, and each step is a piece that fits together to reveal the final picture. This answer, -54, is not just a number; it's the result of a logical journey, a testament to the beauty and precision of mathematics. Understanding the process is just as important as getting the right answer, as it builds a foundation for tackling even more challenging problems in the future. Now, take a moment to appreciate the simplicity and elegance of the solution. This single number encapsulates the interaction of three different functions, showcasing the interconnectedness of mathematical concepts.

Let's Summarize

To recap, we evaluated the composite function (f ∘ h)(x) for x = -1, given f(x) = x³ + 9x, g(x) = √(2x), and h(x) = 8x + 5. We found h(-1) to be -3 and then plugged that into f(x) to get f(-3) = -54. Thus, (f ∘ h)(-1) = -54. It might seem complicated at first, but breaking it down into smaller steps makes it much easier to handle. Composite functions are a crucial concept in mathematics, and mastering them opens the door to more advanced topics. Understanding how functions interact and how to evaluate them sequentially is a valuable skill that extends beyond the classroom. The process of evaluating composite functions teaches us to think logically and methodically, skills that are applicable in many areas of life. So, whether you're working on a math problem or tackling a real-world challenge, remember the power of breaking things down into smaller, manageable steps.

In conclusion, we successfully evaluated the composite function (f ∘ h)(x) at x = -1. Remember to practice these steps, and you'll become a composite function pro in no time! For further learning on composite functions, check out this helpful resource on Khan Academy. Happy math-ing!