Evaluating Composite Functions Finding G(h(10))

In the realm of mathematics, composite functions present an intriguing concept where the output of one function serves as the input for another. This article delves into the process of evaluating composite functions, using the example of g(h(10)) given the functions g(x) = √(x - 4) and h(x) = 2x - 8. We will break down the problem step by step, ensuring a clear understanding of the underlying principles and arriving at the correct solution. This comprehensive guide will not only solve this specific problem but also equip you with the knowledge to tackle similar challenges involving composite functions. Understanding composite functions is crucial for various mathematical applications, including calculus and real-world modeling.

Understanding Composite Functions

Composite functions are formed when one function is plugged into another. In mathematical notation, the composite function of g with h is written as g(h(x)), which means we first evaluate the function h(x) and then use its output as the input for the function g(x). This sequential application of functions is a cornerstone of advanced mathematical concepts. Mastering this concept is essential for understanding more complex topics such as chain rule in calculus and various transformations in geometry. The order of operations is critical when dealing with composite functions; switching the order of g and h can lead to drastically different results. Before diving into our specific problem, let's solidify our understanding with a general example. Consider f(x) = x + 1 and k(x) = x². Then f(k(x)) = f(x²) = x² + 1, while k(f(x)) = k(x + 1) = (x + 1)² = x² + 2x + 1. This simple example illustrates the importance of the order in which functions are composed. Visualizing composite functions can also be helpful. Imagine two machines: one that performs the function h and another that performs the function g. The output from machine h becomes the input for machine g. This visual representation aids in understanding the flow of operations and the dependency between the two functions.

Problem Statement: Evaluating g(h(10))

Our primary goal is to determine the value of g(h(10)), given that g(x) = √(x - 4) and h(x) = 2x - 8. This involves a two-step process: first, we need to evaluate h(10), and then we will use the result as the input for the function g. This process of function composition is a fundamental concept in mathematics and is used extensively in various fields, including computer science, engineering, and physics. Let's break down each step to ensure clarity and accuracy. The problem presented is a typical example of how composite functions are evaluated. By understanding the steps involved, we can apply the same methodology to solve other problems involving different functions and inputs. The ability to evaluate composite functions accurately is a key skill in algebra and pre-calculus, laying the foundation for more advanced topics in calculus and beyond. Moreover, composite functions often appear in real-world applications, such as modeling the effects of a series of transformations or calculations. For instance, in economics, one function might represent the cost of producing a certain number of items, while another function might represent the revenue generated from selling those items. The composite function could then represent the overall profit.

Step 1: Evaluating h(10)

The first step in finding g(h(10)) is to determine the value of h(10). We are given the function h(x) = 2x - 8. To evaluate h(10), we substitute x with 10 in the expression for h(x). Substitution is a fundamental operation in algebra and is crucial for evaluating functions and solving equations. By replacing x with 10, we get h(10) = 2(10) - 8. Now we perform the arithmetic operations: 2(10) = 20, and then 20 - 8 = 12. Therefore, h(10) = 12. This result is the input for the next step, where we will evaluate the function g. It's important to note the order of operations (PEMDAS/BODMAS) when performing these calculations. Multiplication should be done before subtraction. A common mistake is to subtract 8 from 10 first, which would lead to an incorrect result. Understanding the order of operations is essential for accurate mathematical calculations. Moreover, this step highlights the concept of a function as a mapping or transformation. The function h takes the input 10 and transforms it into the output 12. This transformation view of functions is useful in visualizing their behavior and applications. The calculation of h(10) is a straightforward application of the function definition, but it forms a critical building block for evaluating the composite function.

Step 2: Evaluating g(h(10)) = g(12)

Now that we know h(10) = 12, we can substitute this value into the outer function g. We need to find g(12), where g(x) = √(x - 4). Substituting x with 12 in the expression for g(x), we get g(12) = √(12 - 4). First, we perform the subtraction inside the square root: 12 - 4 = 8. So, we have g(12) = √8. To simplify √8, we look for perfect square factors of 8. We can write 8 as 4 * 2, where 4 is a perfect square. Therefore, √8 = √(4 * 2) = √4 * √2 = 2√2. Thus, g(12) = 2√2. This completes the evaluation of the composite function g(h(10)). The simplification of the square root is an important step in obtaining the final answer in its simplest form. Understanding how to factor numbers and identify perfect squares is a valuable skill in algebra. Moreover, the process of simplifying radicals often appears in various mathematical contexts, including geometry and trigonometry. The result 2√2 is an exact value, which is preferable to a decimal approximation in many cases. The use of exact values ensures accuracy in further calculations if the result is used in subsequent steps. This step also demonstrates the importance of understanding the domain of a function. The function g(x) = √(x - 4) is only defined for values of x such that x - 4 ≥ 0, or x ≥ 4. Since 12 is greater than 4, the function is defined at x = 12, and our calculation is valid.

Final Answer and Conclusion

Therefore, we have found that g(h(10)) = 2√2. This corresponds to answer choice A. This solution demonstrates the process of evaluating composite functions by working from the inside out, first evaluating the inner function and then using its result as the input for the outer function. The step-by-step approach ensures accuracy and clarity in the solution. The key takeaway from this problem is the understanding of function composition and how to apply it in practice. Composite functions are a powerful tool in mathematics and have numerous applications in various fields. By mastering the concept of function composition, you can tackle more complex problems and gain a deeper understanding of mathematical relationships. The ability to break down a complex problem into smaller, manageable steps is a valuable problem-solving skill that extends beyond mathematics. In this case, we first evaluated h(10) and then used the result to evaluate g(h(10)). This systematic approach can be applied to a wide range of problems in different domains. Moreover, this example reinforces the importance of understanding the definitions of functions and how to apply them correctly. The functions g(x) and h(x) were defined explicitly, and we used these definitions to evaluate their values at specific points. Overall, this problem provides a solid foundation for understanding and working with composite functions.