Understanding Composite Functions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of composite functions. Specifically, we'll tackle a problem where we need to figure out what the composition of two functions looks like. Don't worry, it's not as scary as it sounds. We'll break it down step by step so you can ace your next math quiz. So, buckle up, grab your favorite beverage, and let's get started! We'll go through this type of math problem, making sure you understand what's going on. In this problem, we are given two functions, h(x) and k(x), and we need to find the expression equivalent to (k ∘ h)(x), which represents a composite function. So, first things first, let's define what a composite function is. It's basically a function within a function. We're taking the output of one function and using it as the input for another function. It's like a mathematical chain reaction.

Let's explore composite functions and the process of evaluating them. In our example, we have two functions: h(x) = 5 + x and k(x) = 1/x. The notation (k ∘ h)(x) means we're going to apply the function h first, and then apply the function k to the result. Think of it like this: the output of h(x) becomes the input for k(x). So, what does this look like in practice? Well, first, we take the function h(x) = 5 + x. This function takes any input x and adds 5 to it. Next, we need to use this whole expression as the input for k(x). The function k(x) = 1/x takes any input and finds its reciprocal. In our case, the input is (5 + x).

So, instead of just having x in the function k, we're going to have (5 + x). It's like plugging one function into the other. This means wherever we see x in k(x), we're going to replace it with (5 + x). Therefore, (k ∘ h)(x) becomes 1/(5 + x). This is the equivalent expression we're looking for. Pretty cool, right? It might seem a bit tricky at first, but with practice, you'll become a composite function master. Always remember the order of operations: work from the inside out. Figure out what the inner function does, and then use that result as the input for the outer function. Keep practicing, and you'll be solving these problems like a pro in no time! Now, let's go through the answer choices to pinpoint the correct one. We're looking for an expression equivalent to 1/(5 + x).

Decoding the Composition: A Detailed Look

Let's break down the composition of functions, (k ∘ h)(x), step-by-step, and meticulously analyze the provided answer choices. This approach ensures a comprehensive understanding and reinforces the concepts of composite functions. The problem presents us with two functions: h(x) = 5 + x and k(x) = 1/x. The core task involves finding the expression that results from applying h(x) first and then using its output as the input for k(x). The notation (k ∘ h)(x) signifies that we must evaluate h(x) and then substitute its result into k(x).

To start, we know that h(x) = 5 + x. This means, for any value of x, h(x) will add 5 to it. Now, we need to find what k(h(x)) is. Given k(x) = 1/x, we replace the x in k(x) with the entire expression of h(x), which is (5 + x). So, we substitute (5 + x) for x in k(x). This yields k(h(x)) = 1/(5 + x). Therefore, the composite function (k ∘ h)(x) is equivalent to 1/(5 + x). Now, let's carefully consider the multiple-choice options provided in the problem. Understanding how to work with composite functions is super helpful, and it's a fundamental concept in algebra. Being able to identify the correct composite function is a crucial skill that will help you in various areas of mathematics. The most important thing is to remember the order of operations.

Examining the Answer Choices

Let's examine the given answer choices to find the equivalent expression for (k ∘ h)(x), which is 1/(5 + x). We'll methodically evaluate each option to determine its correctness. Remember, the goal is to find the option that matches our derived expression.

• A. (5 + x)/x: This option suggests dividing (5 + x) by x. This is incorrect because we are taking the reciprocal of (5 + x), not dividing (5 + x) by x. The correct composition involves putting the output of h(x) as the input of k(x). This option does not represent the correct composition.
• B. 1/(5 + x): This option perfectly matches our derived expression. It indicates taking the reciprocal of (5 + x), which is exactly what (k ∘ h)(x) represents. It correctly substitutes the output of h(x) into k(x), resulting in the correct composite function. So, this is our correct answer!
• C. 5 + (1/x): This option suggests adding 5 to the reciprocal of x. This is incorrect because it represents h(k(x)), the reverse composition. It doesn't align with the order of operations required for (k ∘ h)(x), where we apply h first.
• D. 5 + (5 + x): This option adds (5 + x) to 5. This is completely unrelated to the composition (k ∘ h)(x). It doesn't involve the reciprocal function k(x) at all, nor does it correctly apply the functions in the proper order. Therefore, this option is incorrect.

Therefore, by process of elimination and direct calculation, option B, 1/(5 + x), is the only correct answer. This analysis not only solves the problem but also reinforces the fundamental principles of composite functions. Understanding these principles is a stepping stone to mastering more complex mathematical concepts. It's like building blocks; each concept builds upon the previous one. So, keep up the good work, and you'll be building those mathematical towers in no time.

Summary of the Problem and Solution

In essence, the problem asks us to find the expression for the composite function (k ∘ h)(x), where h(x) = 5 + x and k(x) = 1/x. To solve this, we first found the output of h(x), which is (5 + x). Then, we substituted this output into the function k(x), replacing x with (5 + x). This gave us 1/(5 + x). After carefully examining the answer choices, we identified that option B, 1/(5 + x), correctly represents the composite function (k ∘ h)(x). This confirms our understanding of composite functions and how to work with them.

In this type of question, it's always crucial to correctly identify the order of operations. The order matters! Remember, the function closest to the variable x is applied first. In the case of (k ∘ h)(x), h(x) is applied first. Another common area of confusion is correctly substituting one function into the other. Be careful not to make the mistake of, for example, trying to calculate h(k(x)), which is the inverse. Keep practicing, and you'll find that solving composite function problems becomes second nature.

This method is like detective work – you're looking for clues in the form of function definitions and then carefully applying the rules to solve the mystery! Mastering composite functions opens doors to understanding more advanced topics in calculus and other areas of mathematics. It's a building block for your mathematical journey, so keep practicing and you'll be amazed at how much you grow.

I hope this detailed explanation helps you better understand composite functions. Keep up the great work, and happy calculating!