Solve For (k ∘ H)(x) Given H(x) = 5 + X And K(x) = 1/x

by James Vasile 55 views

Hey there, math enthusiasts! Today, we're diving into the fascinating world of function composition. We've got two functions, h(x) = 5 + x and k(x) = 1/x, and our mission is to figure out which expression is equivalent to (k ∘ h)(x). This notation might look a bit cryptic at first, but don't worry, we'll break it down step by step. Function composition, in simple terms, is like a mathematical assembly line. You feed an input into one function, and then you take the output of that function and feed it into another. In this case, we're feeding 'x' into 'h', and then we're taking the result and feeding it into 'k'. The notation (k ∘ h)(x) specifically means k(h(x)). It tells us to first apply the function 'h' to 'x', and then apply the function 'k' to the result. So, let's get started and unravel this mathematical puzzle!

Dissecting the Functions: h(x) and k(x)

Before we jump into the composition, let's take a closer look at our individual functions. The function h(x) = 5 + x is a straightforward one. It simply takes any input 'x' and adds 5 to it. For instance, if you input 2, the output would be 7. If you input -3, the output would be 2. It's a linear function, meaning its graph would be a straight line. This function is a fundamental building block in many mathematical contexts, often used to represent simple shifts or translations. Understanding its behavior is crucial for grasping more complex concepts. Understanding the linear nature of h(x) is key to visualizing its effect on the input value 'x'. Whether 'x' is positive, negative, or zero, h(x) consistently adds 5, effectively shifting the value along the number line. This characteristic will be important when we consider the composition with k(x). The function k(x) = 1/x is a bit more interesting. It takes any input 'x' (except 0, since division by zero is undefined) and returns its reciprocal. The reciprocal of a number is simply 1 divided by that number. So, the reciprocal of 2 is 1/2, the reciprocal of -4 is -1/4, and so on. This function has a special property: as 'x' gets larger, 1/x gets smaller, and as 'x' gets smaller, 1/x gets larger. This inverse relationship is a hallmark of reciprocal functions. The reciprocal nature of k(x) introduces an element of transformation that is distinct from the simple addition performed by h(x). When we compose k(x) with h(x), this reciprocal effect will play a significant role in the final expression. Think about how the output of h(x) will now become the denominator in k(x), potentially leading to significant changes in the overall behavior of the composite function.

The Heart of the Matter: Composing k(x) and h(x)

Now comes the exciting part: putting these functions together! Remember, (k ∘ h)(x) means we first apply 'h' to 'x', and then apply 'k' to the result. So, the first step is to find h(x), which we already know is 5 + x. This is the inner function, the one that acts on 'x' first. Next, we take this result, (5 + x), and feed it into the function 'k'. The function 'k' takes its input and returns its reciprocal. Therefore, we need to find k(5 + x). Since k(x) = 1/x, then k(5 + x) = 1/(5 + x). We've essentially replaced the 'x' in 'k(x)' with the entire expression (5 + x). This is the core idea behind function composition: substituting one function's output as the input for another. The act of substitution is where the magic of composition happens. We're not just adding or multiplying; we're creating a chain reaction where the output of one function directly influences the input of the next. This creates a new function with its own unique characteristics, shaped by the interplay of the individual functions. In our case, the addition in h(x) combined with the reciprocal in k(x) leads to a function with interesting properties, particularly in terms of its domain and range. This single step of substituting (5 + x) into k(x) is the key to simplifying the composite function. Once we've made this substitution, the rest is just a matter of recognizing the resulting expression among the answer choices.

Evaluating the Options: Which Expression Fits?

Okay, we've determined that (k ∘ h)(x) = 1/(5 + x). Now, let's look at the options and see which one matches our result.

  • A. (5 + x)/x: This expression has (5 + x) in the numerator, which is the opposite of what we need. It's the reciprocal of what we want, but not the correct composition.
  • B. 1/(5 + x): Bingo! This is exactly what we derived. It has 1 in the numerator and (5 + x) in the denominator, perfectly matching our result.
  • C. 5 + (1/x): This expression represents a different composition: (h ∘ k)(x), where we apply 'k' first and then 'h'. It's not what we're looking for in this case.
  • D. 5 + (5 + x): This expression is simply adding the functions together, not composing them. It's a linear function, whereas our composite function is a rational function. Carefully examining each option is crucial in multiple-choice questions. It's easy to get tripped up by expressions that look similar but represent different operations or compositions. By systematically comparing each option with our derived result, we can confidently identify the correct answer. Pay close attention to the order of operations and the placement of terms, as these details can significantly alter the meaning of the expression. In this case, the difference between 1/(5 + x) and (5 + x)/x is a critical distinction.

The Verdict: Option B is the Winner

So, after carefully dissecting the functions, composing them correctly, and evaluating the options, we've arrived at the answer. The expression equivalent to (k ∘ h)(x) is 1/(5 + x), which corresponds to option B. Congratulations, we've successfully navigated the world of function composition! This exercise not only provides the answer to a specific question but also strengthens our understanding of how functions interact and transform inputs. Remember, function composition is a powerful tool in mathematics, allowing us to build complex models from simpler building blocks. Mastering function composition opens doors to understanding more advanced mathematical concepts. It's a fundamental operation that appears in calculus, differential equations, and many other areas of mathematics. By practicing these types of problems, you're building a solid foundation for future mathematical endeavors. Don't be afraid to experiment with different functions and compositions to further solidify your understanding. The more you practice, the more comfortable you'll become with this powerful tool.

Key Takeaways and Final Thoughts

Let's recap the key steps we took to solve this problem. First, we understood the meaning of function composition and the notation (k ∘ h)(x). Then, we analyzed the individual functions, h(x) = 5 + x and k(x) = 1/x, to understand their behavior. Next, we performed the composition, substituting h(x) into k(x) to obtain 1/(5 + x). Finally, we compared our result with the given options and identified the correct answer, which was option B. Reflecting on the problem-solving process is just as important as finding the correct answer. By understanding the steps we took, the reasoning behind each step, and the potential pitfalls, we can improve our problem-solving skills for future challenges. Function composition, like many mathematical concepts, requires a combination of understanding the underlying principles and practicing applying those principles to specific problems. Keep practicing, keep exploring, and keep having fun with math! This journey of mathematical discovery is filled with challenges and rewards, and every problem solved is a step forward in your understanding. Remember, the beauty of mathematics lies in its ability to connect seemingly disparate concepts and reveal elegant solutions to complex problems. The elegance of function composition is that it allows us to create new functions with unique properties by combining existing functions. This ability to build complexity from simplicity is a hallmark of mathematical thinking, and it's a skill that will serve you well in many areas of life, not just in mathematics.