Evaluating Composite Functions G(f(2)) A Step-by-Step Solution

by ADMIN 63 views

In the realm of mathematics, composite functions play a crucial role in connecting different functions and creating more complex relationships. Understanding how to evaluate composite functions is an essential skill for any student delving into algebra and beyond. This article aims to provide a comprehensive guide on evaluating composite functions, using a specific example to illustrate the process. We will break down the steps involved in finding the value of g(f(2)), given the functions f(x) = 4x³ - 10 and g(x) = (3x - 4) / 2. By the end of this guide, you'll be equipped with the knowledge and confidence to tackle similar problems.

Understanding Composite Functions

Before diving into the specific problem, let's first understand what composite functions are. A composite function is essentially a function that is formed by applying one function to the result of another function. In mathematical notation, if we have two functions, f(x) and g(x), then the composite function g(f(x)) represents applying the function f to x first, and then applying the function g to the result. The order of operations is crucial here, as g(f(x)) is generally different from f(g(x)). Visualizing composite functions as a chain of operations can be helpful. The input x goes into the first function, f, and the output of f becomes the input for the second function, g. The final result is the output of the composite function.

Understanding the notation is also key. The notation g(f(x)) is read as "g of f of x." It means we are substituting the entire function f(x) into the variable x of the function g(x). This process might seem abstract at first, but with practice, it becomes a straightforward procedure. Common mistakes arise from misunderstanding the order of operations or incorrectly substituting the functions. It is essential to pay close attention to the parentheses and follow the correct sequence of applying the functions. Composite functions are not just abstract mathematical concepts; they have real-world applications in various fields, including physics, engineering, and computer science. They are used to model complex systems and relationships where one process depends on the outcome of another.

Problem Statement: Finding g(f(2))

Now, let's focus on the specific problem we aim to solve. We are given two functions:

  • f(x) = 4x³ - 10
  • g(x) = (3x - 4) / 2

Our goal is to find the value of g(f(2)). This means we need to first evaluate f(2), which is the inner function, and then use that result as the input for the function g. This step-by-step approach is the key to solving composite function problems. We will start by substituting 2 into the function f(x) and calculating the result. This value will then be substituted into the function g(x) to obtain the final answer. By breaking down the problem into smaller, manageable steps, we can avoid confusion and ensure accuracy. This method is applicable to any composite function evaluation, regardless of the complexity of the individual functions.

Step 1: Evaluating f(2)

The first step in finding g(f(2)) is to evaluate f(2). This means we need to substitute x = 2 into the function f(x) = 4x³ - 10. The substitution process involves replacing every instance of x in the function with the value 2. So, we have:

f(2) = 4(2)³ - 10

Now, we need to simplify this expression following the order of operations (PEMDAS/BODMAS). First, we calculate the exponent: 2³ = 2 * 2 * 2 = 8. Then, we perform the multiplication: 4 * 8 = 32. Finally, we subtract 10:

f(2) = 32 - 10 = 22

Therefore, f(2) = 22. This result is crucial because it becomes the input for the function g in the next step. It is important to double-check the calculations at this stage to ensure accuracy. Any error in evaluating f(2) will propagate through the rest of the solution. Understanding the order of operations and careful substitution are essential skills for this step. The ability to correctly evaluate functions for specific inputs is a foundational skill in algebra and calculus.

Step 2: Evaluating g(f(2))

Now that we have found f(2) = 22, we can proceed to the next step: evaluating g(f(2)). Since f(2) = 22, we are essentially finding g(22). This means we need to substitute x = 22 into the function g(x) = (3x - 4) / 2. Replacing x with 22, we get:

g(22) = (3(22) - 4) / 2

Again, we follow the order of operations to simplify this expression. First, we perform the multiplication: 3 * 22 = 66. Then, we subtract 4: 66 - 4 = 62. Finally, we divide by 2:

g(22) = 62 / 2 = 31

Therefore, g(f(2)) = 31. This is the final answer to the problem. It represents the value of the composite function g(f(x)) when x = 2. It is helpful to review the entire process to ensure that all steps were performed correctly. Double-checking the calculations and the substitution process can help prevent errors. Understanding how to evaluate composite functions is a fundamental skill in mathematics, and this example demonstrates the step-by-step approach to solving such problems.

Final Answer and Conclusion

In conclusion, we have successfully evaluated the composite function g(f(2)) for the given functions f(x) = 4x³ - 10 and g(x) = (3x - 4) / 2. By following a systematic approach, we first found f(2) = 22 and then used this result to calculate g(22) = 31. Therefore, the final answer is g(f(2)) = 31. This corresponds to option D in the multiple-choice options provided.

Key takeaways from this exercise include:

  • Understanding the definition and notation of composite functions.
  • Following the correct order of operations when evaluating expressions.
  • Substituting values accurately into functions.
  • Breaking down complex problems into smaller, manageable steps.

The ability to evaluate composite functions is essential for further studies in mathematics, particularly in calculus and analysis. This example provides a solid foundation for tackling more complex problems involving composite functions. Practice is key to mastering this skill, so try applying this step-by-step approach to other similar problems. Understanding the underlying concepts and practicing different types of problems will build your confidence and proficiency in working with composite functions. Remember to always double-check your work and pay close attention to the details of each step. With consistent practice, you can confidently solve any composite function problem.