Evaluating Composite Functions A Step By Step Guide

by ADMIN 52 views

In the realm of mathematics, composite functions play a crucial role in combining different functions to create new ones. This article delves into the concept of composite functions, specifically focusing on evaluating the composite function (s o t)(-7) given the functions s(x) = 2 - x^2 and t(x) = 3x. We will break down the process step-by-step, ensuring a clear understanding of the underlying principles. The question at hand involves understanding the composition of functions, a fundamental concept in algebra and calculus. We are given two functions, s(x) and t(x), and asked to find the value of their composition (s ∘ t) at x = -7. This means we need to first evaluate t(-7), and then use that result as the input for s(x). This article aims to provide a comprehensive explanation of how to solve this problem, covering the key concepts and steps involved. This concept is essential for students learning algebra and calculus, as it builds a foundation for more advanced topics. Understanding how functions interact and combine is key to solving more complex mathematical problems. By working through this problem, readers will gain a clearer understanding of composite functions and how to evaluate them. This knowledge can be applied to various mathematical scenarios, making it a valuable skill for anyone studying mathematics or related fields. This article not only provides the solution but also emphasizes the importance of understanding the underlying concepts, ensuring that readers can apply this knowledge to different problems.

Composite Functions: A Deep Dive

Before we dive into the specific problem, let's first understand what composite functions are. A composite function is a function that is formed by applying one function to the result of another. In other words, if we have two functions, f(x) and g(x), the composite function f(g(x)), also written as (f o g)(x), means we first apply the function g to x, and then apply the function f to the result. Understanding composite functions is essential for success in various areas of mathematics, including calculus and analysis. They allow us to model complex relationships by combining simpler functions. The order in which we apply the functions is crucial, as f(g(x)) is generally different from g(f(x)). This difference highlights the importance of careful consideration when working with composite functions. Composite functions are not just abstract mathematical concepts; they have real-world applications. For example, they can be used in computer graphics to transform objects, in economics to model supply and demand, and in physics to describe the motion of objects. By understanding composite functions, we can gain a deeper insight into the world around us and solve a wide range of problems. The notation (f o g)(x) represents the composition of f with g, where g is applied first, followed by f. This notation is widely used and provides a concise way to express composite functions. This can be visualized as a chain of operations, where the output of one function becomes the input of the next. Recognizing this chain is key to understanding and evaluating composite functions effectively. Mastering composite functions opens doors to more advanced mathematical concepts and applications, making it a fundamental building block in mathematical education.

Problem Breakdown: Evaluating (s o t)(-7)

Now, let's break down the problem at hand. We are given two functions:

  • s(x) = 2 - x^2
  • t(x) = 3x

We need to find the value of (s o t)(-7), which means we need to find s(t(-7)). To solve this composite function, we must follow a specific order of operations. First, we evaluate the inner function, t(-7), and then we use the result as the input for the outer function, s(x). This step-by-step approach ensures that we correctly apply the functions and arrive at the accurate answer. The first step is to substitute -7 into the function t(x). This gives us t(-7) = 3 * (-7) = -21. This value is the output of the inner function and will be used as the input for the outer function. The next step is to substitute -21 into the function s(x). This gives us s(-21) = 2 - (-21)^2. This is the key step where we combine the results of the two functions. We then need to simplify this expression to find the final value. By carefully following these steps, we can accurately evaluate the composite function and arrive at the correct answer. This process demonstrates the importance of order of operations in mathematics and the need for careful calculation. Understanding the breakdown of this problem provides a clear path to solving similar problems involving composite functions. This approach can be applied to any composite function, regardless of the specific functions involved.

Step-by-Step Solution: Finding s(t(-7))

Let's walk through the solution step-by-step:

  1. Evaluate t(-7):

    Substitute x = -7 into the function t(x) = 3x:

    t(-7) = 3 * (-7) = -21

    This step is crucial because it provides the input for the next function. We are essentially finding the value that the inner function, t(x), produces when the input is -7. This value will then be used as the input for the outer function, s(x). The calculation here is straightforward, involving simple multiplication. However, it is important to pay attention to the signs to ensure accuracy. A mistake in this step will propagate through the rest of the solution, leading to an incorrect answer. Therefore, careful attention to detail is essential in this initial step. The result, -21, is a key intermediate value that bridges the two functions in the composition. This value represents the output of the first function and the input of the second function, illustrating the connection between the two functions in the composite.

  2. Evaluate s(t(-7)) = s(-21):

    Now that we know t(-7) = -21, we substitute this value into the function s(x) = 2 - x^2:

    s(-21) = 2 - (-21)^2

    s(-21) = 2 - 441

    s(-21) = -439

    This step is the heart of the composite function evaluation. We are now using the output of the first function, t(-7), as the input for the second function, s(x). This highlights the sequential nature of composite functions. The calculation involves squaring -21 and then subtracting the result from 2. It is important to remember the order of operations (PEMDAS/BODMAS) and perform the exponentiation before the subtraction. The negative sign in front of the squared term is also crucial to consider carefully. A mistake in squaring or handling the negative sign can lead to an incorrect result. The final result, -439, represents the value of the composite function (s o t)(-7). This value encapsulates the combined effect of the two functions acting sequentially on the input -7. This final calculation demonstrates the power of composite functions in combining different operations into a single mathematical expression.

Therefore, (s o t)(-7) = -439.

Analyzing the Answer Choices

Now that we have found the value of (s o t)(-7) to be -439, we can compare it to the answer choices provided:

A. -439 B. -141 C. 153 D. 443

Clearly, the correct answer is A. -439. Analyzing answer choices is a critical step in problem-solving, especially in multiple-choice questions. It allows us to verify our solution and identify any potential errors. In this case, our calculated value matches one of the answer choices, confirming that our solution is likely correct. However, it's always a good practice to double-check our work to ensure accuracy. We can retrace our steps, review the calculations, and make sure we have not made any mistakes in applying the functions or performing the arithmetic. If the calculated value does not match any of the answer choices, it indicates a potential error in our solution process. In such cases, we need to carefully review our steps and identify the source of the error. Analyzing the answer choices also helps us develop a better understanding of the problem and the range of possible solutions. By considering the different options, we can gain insights into the behavior of the functions and the effect of composition. This analysis can be particularly helpful in more complex problems where the answer is not immediately obvious.

Key Takeaways: Mastering Composite Functions

This problem illustrates the importance of understanding composite functions and how to evaluate them. Here are some key takeaways from this exercise:

  • Order of Operations: When evaluating composite functions, always start with the inner function and work your way outwards.
  • Substitution: Substitute the value obtained from the inner function into the outer function.
  • Careful Calculation: Pay close attention to signs and exponents to avoid errors.
  • Answer Analysis: Compare your answer with the given choices to verify your solution.

These takeaways are crucial for mastering composite functions and solving related problems. Understanding the order of operations is fundamental, as it ensures that we apply the functions in the correct sequence. Substitution is the key mechanism for linking the inner and outer functions, allowing us to combine their effects. Careful calculation is essential to avoid arithmetic errors, which can significantly impact the final result. Finally, answer analysis provides a valuable check on our work and helps us confirm the accuracy of our solution. By internalizing these takeaways, we can confidently approach problems involving composite functions and solve them effectively. These principles are not only applicable to this specific problem but also to a wide range of mathematical scenarios. They provide a solid foundation for understanding and working with functions in various contexts. Mastering these concepts will undoubtedly enhance your mathematical skills and problem-solving abilities.

Conclusion: Applying the Knowledge

In conclusion, evaluating composite functions requires a systematic approach and a clear understanding of the underlying concepts. By following the steps outlined in this article, you can confidently solve similar problems and master the art of function composition. The knowledge gained from this problem can be applied to a wide range of mathematical problems and real-world applications. Composite functions are fundamental in calculus, where they are used to analyze rates of change and optimization problems. They are also essential in computer science, where they are used in programming and algorithm design. Furthermore, composite functions find applications in physics, engineering, and economics, where they help model complex systems and relationships. By mastering composite functions, you are equipping yourself with a powerful tool that can be used in various fields. The ability to understand and manipulate composite functions is a valuable asset in any quantitative discipline. This skill enables you to analyze complex systems, solve intricate problems, and make informed decisions. So, continue practicing and applying your knowledge of composite functions to expand your mathematical horizons and unlock new possibilities.