Unveiling The Composition: Solving For (k ∘ H)(x)

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Hey math enthusiasts! Let's dive into a fun problem involving function composition. The question asks us to find an expression equivalent to (k ∘ h)(x) given that h(x) = 5 + x and k(x) = 1/x. Don't worry, this isn't as scary as it looks. We'll break it down step by step, making sure everyone understands the concept and how to solve it. Ready to roll?

Understanding Function Composition

First things first, what exactly does (k ∘ h)(x) mean? The '∘' symbol signifies function composition. Basically, it means we're applying one function to the result of another function. In this case, we're applying the function k to the output of the function h. Think of it like a chain reaction: you put a value into h, it spits out a new value, and then you plug that new value into k. This concept is fundamental to understanding this type of problem. Remember, the order is crucial! We apply the function closest to the variable x first – in this case, h(x).

Breaking Down the Problem

To solve this, we need to understand how composition works. With h(x) = 5 + x and k(x) = 1/x, we want to find k(h(x)). This means we replace x in the k(x) function with the entire expression of h(x). We are essentially substituting the output of h(x) as the input for k(x). Let's see this in action! Since h(x) = 5 + x, we replace every instance of 'x' in the k(x) equation with the expression '5 + x'. If you can understand the process, you're golden.

Step-by-Step Solution

Alright guys, let's get into the nitty-gritty and work through the solution. This is where we put our understanding to the test and find the equivalent expression. You can totally do this!

  1. Start with the outer function: Our outer function is k(x) = 1/x. We're going to keep this in mind as we start to do the function composition. This is where you replace x in the k(x) equation.
  2. Substitute h(x) into k(x): Since (k ∘ h)(x) means k(h(x)), we substitute h(x) (which is 5 + x) into the place of x in k(x). This gives us k(h(x)) = 1 / (5 + x). We replaced the x in k(x) with the entire expression that defines h(x).
  3. Simplify (if possible): In this case, 1 / (5 + x) is already in its simplest form. There's nothing more we can do to reduce or simplify the expression. We can not simplify the expression further.

So, the expression equivalent to (k ∘ h)(x) is 1 / (5 + x).

Analyzing the Answer Choices

Now that we've found our answer, let's take a look at the given choices and see which one matches our solution. This part is a great way to check your work and make sure you haven't made any mistakes along the way. Checking your answers is crucial to the test-taking process.

A. (5 + x) / x: This expression is not equivalent to our answer, as it represents applying function h to x and then dividing by x, instead of substituting h(x) into k(x). B. 1 / (5 + x): Bingo! This is the expression we derived. It correctly shows the function k being applied to the output of h. C. 5 + (1 / x): This represents applying the functions in the reverse order (h ∘ k)(x), which isn't what the question asked for. This means we substituted x into k(x) and added it to 5. D. 5 + (5 + x): This does not correctly represent the process of function composition as described in the question. This one seems like a distractor as well.

Therefore, the correct answer is B. 1 / (5 + x). Congratulations, you nailed it!

Key Takeaways and Tips

  • Understand the notation: Remember that (k ∘ h)(x) = k(h(x)). The order matters! The function closest to x is applied first.
  • Substitute carefully: Always substitute the entire expression of the inner function into the outer function. Don't just substitute the variable; the entire output of the equation matters.
  • Simplify: After substituting, simplify the expression as much as possible. This makes it easier to compare your answer with the given options.
  • Practice, practice, practice: The more you practice function composition problems, the more comfortable and confident you'll become. Do as many practice questions as you can.

Conclusion

And there you have it! We've successfully navigated the world of function composition and found the equivalent expression for (k ∘ h)(x). Remember, function composition might seem tricky at first, but with practice and a solid understanding of the concepts, you'll be able to tackle these problems with ease. Keep up the awesome work, and keep exploring the amazing world of mathematics! This problem helps to showcase how important it is to remember the definition of the concept itself. Remember that these types of problems are designed to test your knowledge of functions, and can be applied in many situations.

If you have any questions or want to explore more examples, feel free to ask. Keep learning, keep exploring, and keep having fun with math! Happy calculating, everyone! Remember that the key is practice. The more you do, the easier it will become. You can totally do this. Now go out there and conquer those math problems! Cheers!