Evaluating Composite Functions S(x) And T(x) Finding S ∘ T (-7)

In mathematics, understanding composite functions is crucial for solving various problems. A composite function is essentially a function that is applied to the result of another function. This article aims to provide a detailed explanation of how to evaluate a composite function, specifically focusing on the expression (s ∘ t)(-7) where s(x) = 2 - x^2 and t(x) = 3x. We will break down the process step by step, ensuring clarity and comprehension for anyone encountering such problems. Mastering composite functions not only enhances your problem-solving skills but also builds a solid foundation for more advanced mathematical concepts. Let's dive into the intricacies of composite functions and learn how to evaluate them effectively.

What are Composite Functions?

At the heart of understanding $(s ∘ t)(-7)$ lies the concept of composite functions. A composite function is a function that is formed by applying one function to the result of another. In mathematical notation, $(s ∘ t)(x)$ represents the composite function where the function $t$ is applied first, and then the function $s$ is applied to the result. This can be read as "$s$ of $t$ of $x$". The order in which the functions are applied is crucial, as changing the order can lead to different results. For instance, $(s ∘ t)(x)$ is generally different from $(t ∘ s)(x)$. The inner function is the one that is applied first (in this case, $t(x)$), and the outer function is the one that is applied to the result of the inner function (in this case, $s(x)$). Understanding this order is fundamental to correctly evaluating composite functions. The concept of composite functions is widely used in various fields of mathematics, including calculus, algebra, and analysis. It allows us to model complex relationships by breaking them down into simpler, more manageable functions. Recognizing and understanding composite functions is a key skill for anyone studying mathematics, enabling them to tackle more advanced problems with confidence. Composite functions also have practical applications in computer science, engineering, and other disciplines, where they are used to model systems and processes that involve multiple steps or transformations. So, mastering composite functions is not just about understanding a mathematical concept; it's about acquiring a powerful tool for problem-solving in a wide range of contexts.

Breaking Down (s ∘ t)(-7)

To effectively evaluate $(s ∘ t)(-7)$, we need to break down the expression into manageable steps. The notation $(s ∘ t)(-7)$ signifies that we first apply the function $t$ to the value -7, and then we apply the function $s$ to the result. This sequential application is the key to understanding composite functions. The first step is to evaluate $t(-7)$. Given that $t(x) = 3x$, we substitute -7 for $x$: $t(-7) = 3 * (-7) = -21$. This result, -21, now becomes the input for the function $s$. The next step is to evaluate $s(-21)$. Since $s(x) = 2 - x^2$, we substitute -21 for $x$: $s(-21) = 2 - (-21)^2 = 2 - 441$. The final step is to perform the subtraction: $2 - 441 = -439$. Therefore, $(s ∘ t)(-7) = -439$. This step-by-step approach ensures that we follow the correct order of operations and arrive at the accurate answer. By breaking down the composite function into its individual components, we can avoid confusion and simplify the evaluation process. This method is applicable to any composite function, regardless of the complexity of the individual functions involved. Understanding this process is crucial for mastering composite functions and applying them effectively in various mathematical contexts. So, remember to always start with the inner function, evaluate its result, and then use that result as the input for the outer function. This systematic approach will help you navigate composite functions with ease and confidence.

Step-by-Step Evaluation

Let's delve into a step-by-step evaluation of $(s ∘ t)(-7)$, reinforcing the process and ensuring a clear understanding. As previously established, we have two functions: $s(x) = 2 - x^2$ and $t(x) = 3x$. The composite function $(s ∘ t)(-7)$ requires us to first apply the function $t$ to -7 and then apply the function $s$ to the result. Step 1: Evaluate t(-7). We substitute -7 into the function $t(x)$: $t(-7) = 3 * (-7) = -21$. This calculation is straightforward: multiplying 3 by -7 gives us -21. This result is crucial as it becomes the input for the next step. Step 2: Evaluate s(t(-7)), which is s(-21). Now we take the result from Step 1, -21, and substitute it into the function $s(x)$: $s(-21) = 2 - (-21)^2$. This step involves squaring -21, which gives us 441. Step 3: Complete the calculation for s(-21). We now have: $s(-21) = 2 - 441$. Subtracting 441 from 2 gives us -439. Therefore, $(s ∘ t)(-7) = -439$. This step-by-step breakdown illustrates the systematic approach required for evaluating composite functions. By breaking the problem into smaller, manageable steps, we can minimize errors and ensure accuracy. This method is applicable to any composite function, regardless of the complexity of the individual functions involved. So, remember to always start with the inner function, evaluate its result, and then use that result as the input for the outer function. This systematic approach will help you navigate composite functions with ease and confidence. The key takeaway is that composite functions are evaluated from the inside out, ensuring that the correct order of operations is followed.

Detailed Solution

To provide a detailed solution for evaluating $(s ∘ t)(-7)$, let's reiterate the functions and the composite function notation. We have $s(x) = 2 - x^2$ and $t(x) = 3x$. The expression $(s ∘ t)(-7)$ means we need to find $s(t(-7))$. This notation emphasizes that we first evaluate the inner function, $t(-7)$, and then use that result as the input for the outer function, $s(x)$. First, let's evaluate t(-7): We substitute -7 into the function $t(x)$: $t(-7) = 3 * (-7)$. Multiplying 3 by -7, we get: $t(-7) = -21$. Now, let's evaluate s(t(-7)), which is s(-21): We substitute -21 into the function $s(x)$: $s(-21) = 2 - (-21)^2$. Next, we square -21: $(-21)^2 = 441$. Now, we substitute this back into the equation: $s(-21) = 2 - 441$. Finally, we subtract 441 from 2: $s(-21) = -439$. Therefore, $(s ∘ t)(-7) = -439$. This detailed solution breaks down each step, making it clear how we arrive at the final answer. The key is to follow the order of operations, starting with the inner function and working outwards. This approach is consistent for all composite function evaluations. By understanding each step and the logic behind it, you can confidently tackle similar problems. Remember, the composite function notation $(s ∘ t)(x)$ represents a sequential application of functions, where $t(x)$ is applied first, and then $s(x)$ is applied to the result. This understanding is crucial for correctly evaluating composite functions.

Identifying the Correct Answer

After performing the detailed evaluation of $(s ∘ t)(-7)$, it's crucial to identify the correct answer from the given options. We have determined that $(s ∘ t)(-7) = -439$. Now, let's look at the options provided: A. -439 B. -141 C. 153 D. 443 By comparing our result with the options, we can clearly see that Option A, -439, is the correct answer. The other options (-141, 153, and 443) do not match our calculated value. This step is essential to ensure that we select the correct answer and avoid any careless mistakes. It also reinforces the importance of accurate calculations and a thorough understanding of the problem. In this case, our step-by-step evaluation has led us to the correct answer, demonstrating the effectiveness of breaking down the problem into smaller, manageable steps. Identifying the correct answer is not just about matching a number; it's about confirming that our entire process, from understanding the problem to performing the calculations, has been accurate. This final step provides confidence in our solution and ensures that we have correctly applied the concepts of composite functions. So, always take the time to compare your result with the given options and verify that you have selected the correct answer. This practice will help you build accuracy and confidence in your problem-solving skills.

Conclusion

In conclusion, evaluating composite functions like $(s ∘ t)(-7)$ involves a systematic approach of applying functions in a specific order. We have seen how to break down the problem into smaller steps, starting with the inner function and working outwards. Given $s(x) = 2 - x^2$ and $t(x) = 3x$, we first evaluated $t(-7)$, which gave us -21. Then, we used this result as the input for $s(x)$, evaluating $s(-21)$, which resulted in -439. Therefore, $(s ∘ t)(-7) = -439$. This process highlights the importance of understanding the notation and the order of operations in composite functions. By following a step-by-step approach, we can accurately evaluate complex expressions and arrive at the correct answer. Mastering composite functions is a valuable skill in mathematics, providing a foundation for more advanced concepts. The ability to break down complex problems into smaller, manageable steps is a key problem-solving strategy that extends beyond mathematics. It's a skill that can be applied in various fields and aspects of life. So, by understanding and practicing composite function evaluations, you are not only enhancing your mathematical skills but also developing valuable analytical and problem-solving abilities. Remember to always start with the inner function, evaluate its result, and then use that result as the input for the outer function. This systematic approach will help you navigate composite functions with ease and confidence.