Evaluating The Composite Function F(g(1)) A Step-by-Step Solution

In this article, we will walk through the process of evaluating a composite function. Specifically, we will focus on the functions f(x) = 3√(x) + 1 and g(x) = x² - 8, and determine the value of f(g(1)). Composite functions might seem daunting at first, but by breaking down the process step-by-step, we can solve this problem methodically. This detailed explanation aims to provide a clear understanding of how to tackle such problems, which often appear in mathematics courses and standardized tests. By the end of this article, you will have a strong grasp on evaluating composite functions, which involves substituting one function into another. This is a fundamental concept in algebra and calculus, making it crucial for anyone studying mathematics. Let’s dive in and demystify the process together!

Understanding Composite Functions

Before we dive into solving for f(g(1)), let's first clarify what a composite function is. A composite function is essentially a function that is applied to the result of another function. In mathematical notation, if we have two functions, f(x) and g(x), the composite function f(g(x)) means we first apply the function g to x, and then apply the function f to the result. The order here is critical; f(g(x)) is generally not the same as g(f(x)). Understanding this order is essential for accurately evaluating composite functions. Many real-world scenarios can be modeled using composite functions. For example, consider a store offering a discount on an item, followed by a sales tax. The final price you pay is a composite function where the discount function is applied first, and then the sales tax function is applied to the discounted price. This concept is also widely used in computer science, where functions are often nested within each other to perform complex operations. To fully grasp this concept, it's beneficial to practice with various examples and understand the underlying principles of function composition. This foundational knowledge will not only help you solve mathematical problems but also appreciate how functions interact in various contexts. Therefore, before moving forward, ensure you are comfortable with the idea of functions being plugged into one another, as this is the key to mastering composite functions.

Step-by-Step Evaluation of f(g(1))

To evaluate f(g(1)), we follow a two-step process, which will ensure we arrive at the correct answer. These steps involve working from the inside out, a common strategy when dealing with composite functions. This approach helps to break down the problem into manageable parts, reducing the likelihood of errors. The first step involves evaluating the inner function, g(x), at x = 1. Once we have the value of g(1), we can then use that result as the input for the outer function, f(x). This method not only simplifies the process but also highlights the sequential nature of composite functions. It's akin to following a recipe where each step must be completed before moving on to the next. Moreover, understanding this process is crucial for tackling more complex composite functions involving multiple layers or different types of functions. So, let’s proceed with the first step, which sets the foundation for our final answer. Remember, accuracy at each step is paramount to ensure the final result is correct. The step-by-step approach not only aids in solving the problem but also reinforces the understanding of how functions interact within a composite structure. By meticulously following these steps, you'll develop a solid foundation in evaluating composite functions, making you more confident in tackling similar problems in the future.

Step 1: Evaluate g(1)

The first step in finding f(g(1)) is to evaluate g(1). Recall that the function g(x) is defined as g(x) = x² - 8. To find g(1), we simply substitute x = 1 into the equation. This substitution is a fundamental operation in algebra, where we replace a variable with its given value to determine the function's output. It's crucial to perform this substitution accurately, as an error here will propagate through the rest of the calculation. So, we have g(1) = (1)² - 8. Now, we simplify the expression by performing the arithmetic operations. First, we calculate 1², which equals 1. Then, we subtract 8 from the result: 1 - 8 = -7. Therefore, g(1) = -7. This result is a critical intermediate value that we will use in the next step. Ensuring the correctness of this value is paramount, as it serves as the input for the outer function f(x). By carefully performing the substitution and arithmetic, we have successfully completed the first step in evaluating the composite function. This methodical approach is key to avoiding errors and building confidence in your ability to handle composite functions. With g(1) now determined, we are well-prepared to move on to the next step, where we will use this value to evaluate f(x).

Step 2: Evaluate f(g(1))

Now that we have found g(1) = -7, we can proceed to the second step: evaluating f(g(1)), which is equivalent to evaluating f(-7). Recall that the function f(x) is defined as f(x) = 3√(x) + 1. To find f(-7), we substitute x = -7 into this equation. This step is similar to the substitution we performed in the previous step, but now we are applying it to a different function and a different input value. The accuracy of this substitution is just as crucial as it was before, as it directly impacts the final result. So, we have f(-7) = 3√(-7) + 1. Now, we encounter a square root of a negative number, which introduces us to the realm of imaginary numbers. The square root of -7 can be written as √(-7) = √(7 * -1) = √7 * √(-1). By definition, √(-1) is represented by the imaginary unit i. Thus, √(-7) = √7 * i. Now, we substitute this back into our equation: f(-7) = 3 * √7 * i + 1. We can rewrite this in the standard form for complex numbers, which is a + bi, where a and b are real numbers and i is the imaginary unit. So, f(-7) = 1 + 3√7 * i. This is the final value of f(g(1)). By carefully handling the imaginary unit and following the order of operations, we have successfully evaluated the composite function. This process highlights the importance of understanding not only function composition but also the properties of different types of numbers, including imaginary numbers.

The Final Answer and Its Significance

After carefully following the steps of evaluating the composite function, we have arrived at the final answer: f(g(1)) = 1 + 3√7 * i. This result is a complex number, which consists of a real part (1) and an imaginary part (3√7 * i). This outcome underscores the importance of being comfortable with complex numbers when working with functions, particularly when dealing with square roots of negative numbers. The significance of this answer extends beyond the specific problem. It illustrates how composite functions can lead to results that are not immediately obvious from the original functions. In this case, even though both f(x) and g(x) are defined for real numbers, the composite function f(g(1)) resulted in a complex number. This is a crucial concept in mathematics, highlighting the rich and sometimes unexpected behavior of functions. Furthermore, this example demonstrates the practical application of imaginary numbers, which are often used in various fields such as electrical engineering and quantum mechanics. The ability to work with complex numbers and understand their role in function evaluation is a valuable skill in advanced mathematics and related disciplines. Therefore, this final answer not only solves the problem at hand but also provides a deeper understanding of the interplay between functions and different number systems. By mastering these concepts, you will be well-equipped to tackle more complex mathematical challenges.

Conclusion: Mastering Composite Functions

In conclusion, we have successfully evaluated the composite function f(g(1)) where f(x) = 3√(x) + 1 and g(x) = x² - 8. By breaking down the problem into manageable steps, first evaluating g(1) and then using that result to evaluate f(g(1)), we arrived at the solution f(g(1)) = 1 + 3√7 * i. This process underscores the importance of a methodical approach when dealing with composite functions. Starting with the inner function and working outwards is a strategy that can be applied to a wide range of similar problems. Moreover, this exercise highlighted the significance of understanding different types of numbers, including real and imaginary numbers, and how they interact within functions. The appearance of the imaginary unit i in our final answer serves as a reminder that function evaluation can sometimes lead to results that extend beyond the realm of real numbers. Mastering composite functions is a critical skill in mathematics, as it forms the basis for more advanced concepts in calculus and analysis. The ability to evaluate composite functions accurately and efficiently is not only essential for academic success but also for practical applications in various fields. By practicing with different examples and understanding the underlying principles, you can build confidence in your ability to tackle complex mathematical problems. This journey through evaluating f(g(1)) has provided valuable insights into the world of functions and their interactions, setting a strong foundation for future mathematical explorations.

Final Answer: The correct answer is A. 3√7i + 1