Question 9 Evaluate G(f(h(3))) Composite Functions Deconstructed

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In mathematics, composite functions represent a fascinating concept where one function acts upon the result of another. This article delves into the process of evaluating composite functions, using the specific example of g(f(h(3))). We will break down the problem step-by-step, providing a clear and comprehensive guide to understanding this important mathematical concept. Mastering composite functions is crucial for success in calculus and other advanced mathematical fields. This article aims to provide a clear and concise explanation of how to evaluate composite functions, focusing on the specific example of g(f(h(3))). By understanding the underlying principles and following the step-by-step approach outlined in this article, you will gain confidence in tackling similar problems and develop a strong foundation in this essential mathematical concept. The principles discussed here will also empower you to solve more intricate problems involving multiple composite functions and different types of mathematical expressions.

Understanding Composite Functions

Before diving into the solution, let's define what composite functions are. A composite function is essentially a function within a function. We apply one function to a variable (say, x), and then we apply another function to the result of the first function. This process can be extended to multiple functions, creating a chain of operations. The notation g(f(x)) signifies that we first apply the function f to x, and then we apply the function g to the result. The order of operations is crucial in composite functions. We always work from the inside out, starting with the innermost function and progressing outwards. This order is paramount to obtaining the correct result. Imagine it as a series of transformations: the input x undergoes its first transformation by f, and the result of that transformation is then transformed again by g. The order ensures that the correct transformations are applied in the proper sequence. In this article, we will demystify this concept and demonstrate its practical application. Understanding composite functions is fundamental to many areas of mathematics, including calculus, where derivatives and integrals of composite functions are frequently encountered.

Problem Statement: g(f(h(3)))

We are given three functions:

  • f(x) = x² + 4
  • g(x) = √(x + 3)
  • h(x) = 2x + 1

Our goal is to evaluate g(f(h(3))). This means we need to find the value of the composite function when x = 3. To solve this, we will follow a step-by-step approach, working from the innermost function outwards. First, we will evaluate h(3), then we will use the result to evaluate f(h(3)), and finally, we will use that result to evaluate g(f(h(3))). This methodical approach ensures accuracy and clarity in the solution process. Each step builds upon the previous one, leading us to the final answer. By breaking down the problem into smaller, manageable steps, we can avoid confusion and minimize the chance of errors. This approach also allows us to better understand the flow of operations and the role of each function in the composition. This problem serves as an excellent example to illustrate the application of composite functions in a practical context.

Step 1: Evaluating h(3)

We start with the innermost function, h(x). To find h(3), we substitute x = 3 into the expression for h(x):

h(3) = 2(3) + 1 = 6 + 1 = 7

Therefore, h(3) = 7. This is the first piece of the puzzle. Now we have a numerical value that we can use as input for the next function. Substituting values into functions is a fundamental skill in mathematics, and this step demonstrates its importance in the context of composite functions. By replacing the variable x with the specific value of 3, we transform the function expression into a numerical calculation. This step is crucial because it allows us to progress to the next stage of the evaluation. The result, h(3) = 7, will now serve as the input for the function f in the next step. This process of sequential evaluation is the essence of working with composite functions.

Step 2: Evaluating f(h(3))

Now that we know h(3) = 7, we can evaluate f(h(3)), which is the same as f(7). We substitute x = 7 into the expression for f(x):

f(7) = 7² + 4 = 49 + 4 = 53

So, f(h(3)) = 53. We have now evaluated the second layer of the composite function. The output of h(3) becomes the input for f(x), demonstrating the chained nature of composite functions. This step highlights the interconnectedness of the functions involved. The result of the previous step, h(3) = 7, is not just an intermediate value; it is the key to unlocking the next stage of the evaluation. By substituting this value into f(x), we are essentially applying the transformation defined by f to the output of h. The calculation f(7) = 53 represents this transformation, and this value will now be used as the input for the final function, g(x).

Step 3: Evaluating g(f(h(3)))

Finally, we can evaluate g(f(h(3))), which is the same as g(53). We substitute x = 53 into the expression for g(x):

g(53) = √(53 + 3) = √56

We can simplify √56 by factoring out the largest perfect square:

√56 = √(4 * 14) = √4 * √14 = 2√14

Therefore, g(f(h(3))) = 2√14. This is the final result of our evaluation. We have successfully navigated through the composite function, applying each function in the correct order and arriving at the solution. The final step involves substituting the result of the previous step, f(h(3)) = 53, into the function g(x). This demonstrates the final transformation in the chain of operations. The simplification of the square root is an important aspect of presenting the answer in its most concise form. By factoring out the perfect square, we reduce the radical to its simplest expression. The final answer, 2√14, represents the value of the composite function g(f(h(3))) when x = 3.

Conclusion

By following these steps, we have successfully evaluated the composite function g(f(h(3))). The key to solving composite functions is to work from the inside out, evaluating each function in turn and using the result as the input for the next function. This step-by-step approach ensures accuracy and clarity in the solution process. Understanding composite functions is a fundamental concept in mathematics, and this example provides a practical illustration of how to apply this concept. We have demonstrated the process of evaluating a composite function by breaking it down into smaller, manageable steps. This approach can be applied to more complex composite functions with any number of nested functions. The ability to work with composite functions is crucial for success in advanced mathematics, and this article has provided a solid foundation for further exploration of this topic. Remember to practice these steps with various functions to solidify your understanding and build your problem-solving skills. Composite functions are a cornerstone of mathematical analysis, and mastering them will greatly enhance your mathematical capabilities.

Final Answer:

The correct answer is 2√14, which corresponds to the option 2√14.