Finding The Value Of G(f(x)) A Step By Step Guide

In the realm of mathematics, functions serve as fundamental building blocks, mapping inputs to corresponding outputs. The composition of functions introduces a fascinating layer of complexity, where the output of one function becomes the input of another. This intricate dance between functions allows us to model complex relationships and explore the interconnectedness of mathematical expressions. In this article, we will embark on a journey to unravel the concept of function composition, specifically focusing on the evaluation of g(f(x)) given two distinct functions, f(x) and g(x). This exploration will not only enhance our understanding of function composition but also equip us with the tools to tackle similar mathematical challenges with confidence. Our primary objective is to determine the precise value of g(f(x)) when f(x) = (1/3)x^2 + 4 and g(x) = 9x - 12. By meticulously substituting f(x) into g(x) and simplifying the resulting expression, we will arrive at the correct answer, demystifying the process of function composition along the way. Understanding function composition is crucial for various mathematical applications, including calculus, where it plays a vital role in chain rule differentiation. Furthermore, it is a fundamental concept in computer science, where functions are extensively used to model algorithms and data transformations. By mastering function composition, we unlock a powerful toolset for solving complex problems and gaining a deeper appreciation for the elegance of mathematical relationships.

Problem Statement: Decoding the Composition

Let us consider the functions f(x) = (1/3)x^2 + 4 and g(x) = 9x - 12. Our mission is to determine the value of g(f(x)), which represents the composition of the functions g and f. To achieve this, we must first understand the essence of function composition. Function composition, denoted as g(f(x)), involves substituting the entire function f(x) into the function g(x) wherever the variable x appears. In simpler terms, we are essentially plugging the output of f(x) into g(x) as its input. This process requires careful substitution and meticulous simplification to arrive at the correct expression for g(f(x)). The given functions, f(x) and g(x), are both relatively straightforward, making this problem an excellent starting point for understanding function composition. f(x) is a quadratic function, characterized by its parabolic shape, while g(x) is a linear function, represented by a straight line. The composition of these two functions will result in a new function that exhibits a combination of their properties. Before we proceed with the step-by-step solution, it is beneficial to have a clear roadmap in mind. We will first substitute f(x) into g(x), then expand the resulting expression, and finally simplify it to obtain the final answer. This methodical approach will minimize the chances of errors and ensure a clear and concise solution. Furthermore, we will discuss the importance of function composition in various mathematical contexts, highlighting its applications in calculus, computer science, and other fields. By understanding the underlying principles and techniques involved in function composition, we can confidently tackle more complex problems and gain a deeper appreciation for the interconnectedness of mathematical concepts.

Step-by-Step Solution: Unraveling the Composition

1. Substitution: The first step in evaluating g(f(x)) is to substitute the expression for f(x) into g(x). This means replacing every instance of 'x' in g(x) with the entire expression for f(x), which is (1/3)x^2 + 4. Mathematically, this can be represented as: g(f(x)) = g((1/3)x^2 + 4). Now, we replace the 'x' in g(x) = 9x - 12 with the expression (1/3)x^2 + 4. This gives us: g(f(x)) = 9((1/3)x^2 + 4) - 12. This substitution is the cornerstone of function composition, and it is crucial to perform it accurately to avoid errors in the subsequent steps. It is helpful to visualize this substitution as plugging one function directly into another, where the output of the inner function becomes the input of the outer function. In this case, the output of f(x) becomes the input of g(x). This substitution process may seem straightforward, but it is essential to pay close attention to the order of operations and the placement of parentheses to ensure that the expression is correctly represented. A minor error in substitution can lead to a completely different result, highlighting the importance of meticulous attention to detail. Furthermore, understanding the concept of substitution is not only crucial for function composition but also for other areas of mathematics, such as solving equations and evaluating expressions. By mastering this fundamental skill, we lay a solid foundation for more advanced mathematical concepts.

2. Expansion: Next, we need to expand the expression obtained in the previous step. This involves distributing the 9 across the terms inside the parentheses: g(f(x)) = 9 * (1/3)x^2 + 9 * 4 - 12. Multiplying 9 by (1/3)x^2 gives us 3x^2, and multiplying 9 by 4 gives us 36. Therefore, the expression becomes: g(f(x)) = 3x^2 + 36 - 12. This expansion step is a crucial application of the distributive property, which states that a(b + c) = ab + ac. The distributive property allows us to simplify expressions involving parentheses and is a fundamental tool in algebra. In this case, we are distributing the constant 9 over the terms (1/3)x^2 and 4. The expansion step not only simplifies the expression but also reveals the individual terms that will contribute to the final result. By expanding the expression, we make it easier to identify like terms that can be combined in the next step. Furthermore, the expansion step helps to clarify the structure of the expression and reveals the relationship between the different terms. This understanding is essential for simplifying the expression and arriving at the correct answer. It is important to perform the expansion step carefully, paying attention to the signs of the terms and ensuring that each term is correctly multiplied. A minor error in expansion can lead to an incorrect result, emphasizing the importance of precision and attention to detail.

3. Simplification: The final step is to simplify the expanded expression by combining like terms. In this case, we have two constant terms, 36 and -12, which can be combined: g(f(x)) = 3x^2 + (36 - 12). Subtracting 12 from 36 gives us 24. Therefore, the simplified expression is: g(f(x)) = 3x^2 + 24. This simplification step is a fundamental algebraic technique that involves combining terms with the same variable and exponent. In this case, we are combining the constant terms 36 and -12. Combining like terms allows us to express the expression in its most concise and simplified form. This simplification not only makes the expression easier to understand but also reduces the chances of errors in subsequent calculations. The simplified expression, g(f(x)) = 3x^2 + 24, represents the final result of the function composition. It is a quadratic function, reflecting the influence of the quadratic function f(x). The constant term 24 represents the vertical shift of the parabola, while the coefficient 3 affects its width and direction. By simplifying the expression, we have successfully evaluated g(f(x)) and obtained a clear and concise representation of the composed function. This simplified expression can be used for further analysis, such as finding the roots of the function or determining its maximum or minimum value. The simplification step is an essential part of the problem-solving process, and it demonstrates the power of algebraic techniques in unraveling complex mathematical expressions.

Final Answer: The Value of g(f(x))

Therefore, the value of g(f(x)) is 3x^2 + 24, which corresponds to option C. This final answer represents the culmination of our step-by-step solution, where we meticulously substituted, expanded, and simplified the expression to arrive at the correct result. The process of function composition may initially seem daunting, but by breaking it down into manageable steps, we can approach it with confidence and clarity. Understanding function composition is not only essential for solving mathematical problems but also for gaining a deeper appreciation for the interconnectedness of mathematical concepts. The final answer, 3x^2 + 24, is a quadratic function, indicating that the composition of a linear function (g(x)) with a quadratic function (f(x)) can result in a quadratic function. This observation highlights the importance of understanding the properties of different types of functions and how they interact with each other under composition. Furthermore, the final answer can be used to explore various aspects of the composed function, such as its graph, its roots, and its behavior as x approaches infinity. This analysis can provide valuable insights into the relationship between the original functions and their composition. In conclusion, the value of g(f(x)) is 3x^2 + 24, and this result demonstrates the power and elegance of function composition in mathematics. By mastering this concept, we can unlock a new level of understanding and appreciation for the beauty of mathematical relationships. The journey of solving this problem has not only provided us with the answer but also equipped us with valuable skills and insights that can be applied to a wide range of mathematical challenges.

Discussion: Exploring the Significance of Function Composition

Function composition is a fundamental concept in mathematics with far-reaching implications. It allows us to build complex functions from simpler ones, creating a hierarchical structure that mirrors the complexity of the real world. In calculus, function composition plays a crucial role in the chain rule, which allows us to differentiate composite functions. The chain rule states that the derivative of g(f(x)) is g'(f(x)) * f'(x), where g' and f' denote the derivatives of g and f, respectively. This rule is essential for differentiating a wide range of functions, including trigonometric functions, exponential functions, and logarithmic functions. Without the chain rule, calculus would be significantly limited in its ability to handle complex functions. In computer science, function composition is a cornerstone of functional programming. Functional programming emphasizes the use of functions as first-class citizens, meaning that functions can be passed as arguments to other functions and returned as values. Function composition allows us to combine smaller, more manageable functions into larger, more complex ones, promoting code reusability and modularity. This approach leads to more maintainable and testable code, as the individual functions are easier to understand and debug. Furthermore, function composition is closely related to the concept of pipelines, where the output of one function is fed into the input of another, creating a chain of transformations. Pipelines are widely used in data processing and analysis, where data is often transformed through a series of steps. Beyond calculus and computer science, function composition finds applications in various other fields, including physics, engineering, and economics. In physics, function composition can be used to model the motion of objects under the influence of multiple forces. In engineering, it can be used to design complex systems by combining simpler components. In economics, it can be used to model the behavior of markets and the interactions between different economic agents. The significance of function composition lies in its ability to create complex models from simpler building blocks. This approach allows us to tackle intricate problems by breaking them down into smaller, more manageable parts. Function composition is a powerful tool for understanding and modeling the world around us, and its applications continue to expand as new fields emerge and existing fields evolve. By mastering this concept, we equip ourselves with a valuable tool for problem-solving and critical thinking.

Conclusion: Mastering the Art of Function Composition

In conclusion, we have successfully determined the value of g(f(x)) for the given functions f(x) = (1/3)x^2 + 4 and g(x) = 9x - 12. Through a meticulous step-by-step process of substitution, expansion, and simplification, we arrived at the final answer: g(f(x)) = 3x^2 + 24. This exploration has not only provided us with the solution to this specific problem but has also deepened our understanding of function composition, a fundamental concept in mathematics. Function composition allows us to combine functions in a meaningful way, creating new functions with unique properties and behaviors. This process is essential for modeling complex relationships and solving a wide range of mathematical problems. We have also discussed the significance of function composition in various fields, including calculus, computer science, physics, engineering, and economics. In calculus, function composition is crucial for the chain rule, which allows us to differentiate composite functions. In computer science, it is a cornerstone of functional programming, promoting code reusability and modularity. In physics, engineering, and economics, function composition can be used to model complex systems and interactions. By mastering the art of function composition, we equip ourselves with a powerful tool for problem-solving and critical thinking. This concept is not merely a theoretical exercise but has practical applications in various domains, making it an essential skill for students, researchers, and professionals alike. As we continue our mathematical journey, the understanding of function composition will serve as a solid foundation for tackling more advanced concepts and challenges. The ability to combine functions and analyze their interactions is a testament to the elegance and power of mathematics. By embracing this concept, we unlock a new level of understanding and appreciation for the beauty of mathematical relationships. The journey of learning mathematics is a continuous process, and function composition is just one of the many milestones along the way. By mastering this concept, we are better prepared to tackle future challenges and contribute to the ever-evolving world of mathematics.