Finding Quotient Functions And Evaluation Of F(x) = -x^2 + 2x - 1 And G(x) = X - 1

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Introduction

In the realm of mathematical functions, understanding how to combine and manipulate them is crucial. One such operation is finding the quotient of two functions, denoted as (f/g)(x). This article delves into the process of determining (f/g)(x) for the functions f(x) = -x^2 + 2x - 1 and g(x) = x - 1, and subsequently evaluating (f/g)(1). This exploration will not only enhance your understanding of function operations but also highlight the importance of considering the domain of the resulting function. In this article, we will explore the fascinating world of function operations, specifically focusing on finding the quotient of two given functions. Our primary goal is to determine the expression for (f/g)(x), where f(x) = -x^2 + 2x - 1 and g(x) = x - 1. Furthermore, we will evaluate this quotient at x = 1, which will lead us to a deeper understanding of the function's behavior and domain. The process involves algebraic manipulation, factoring, and careful consideration of potential singularities. Understanding these concepts is vital for a strong foundation in calculus and advanced mathematical topics. We will also emphasize the importance of identifying any restrictions on the domain of the resulting function, ensuring a comprehensive analysis. This article will provide a step-by-step guide, making it accessible to learners of all levels. By the end, you will be well-equipped to tackle similar problems and appreciate the beauty and elegance of mathematical functions and their interactions. Join us on this exciting journey as we unravel the intricacies of function division and explore the significance of domain restrictions. This exploration will not only solidify your understanding of function operations but also equip you with valuable problem-solving skills applicable in various mathematical contexts. Our journey begins with a clear statement of the problem and will progress through the necessary steps to arrive at a solution, highlighting key concepts along the way. The ultimate aim is to empower you with the ability to confidently handle similar problems and appreciate the nuanced nature of function manipulations.

Finding (f/g)(x)

To find (f/g)(x), we need to divide the function f(x) by the function g(x). This means we have:

(f/g)(x) = f(x) / g(x) = (-x^2 + 2x - 1) / (x - 1)

Now, let's simplify the expression. Notice that the numerator is a quadratic expression that can be factored. Specifically, -x^2 + 2x - 1 is a perfect square trinomial. Factoring it, we get:

-x^2 + 2x - 1 = -(x^2 - 2x + 1) = -(x - 1)^2

Substituting this back into our expression for (f/g)(x), we have:

(f/g)(x) = -(x - 1)^2 / (x - 1)

We can now simplify this expression by canceling out the common factor of (x - 1) in the numerator and the denominator, but we must be cautious. Canceling this factor is valid only when x ≠ 1 because division by zero is undefined. Therefore, for x ≠ 1:

(f/g)(x) = -(x - 1)

So, (f/g)(x) = -x + 1, provided that x ≠ 1. Finding the quotient of functions involves dividing one function by another, but it's crucial to consider the domain of the resulting function. In our case, we have f(x) = -x^2 + 2x - 1 and g(x) = x - 1. To find (f/g)(x), we divide f(x) by g(x): (f/g)(x) = f(x) / g(x) = (-x^2 + 2x - 1) / (x - 1). The next step is to simplify this expression, if possible. We observe that the numerator, -x^2 + 2x - 1, can be factored. Factoring the numerator is a critical step because it allows us to identify potential common factors with the denominator. In this case, -x^2 + 2x - 1 can be rewritten as -(x^2 - 2x + 1). Recognizing the expression inside the parentheses as a perfect square trinomial is key. We can further factor it as -(x - 1)^2. This factorization is crucial because it reveals a common factor of (x - 1) with the denominator. Substituting the factored form back into our expression, we have (f/g)(x) = -(x - 1)^2 / (x - 1). Now, we can simplify the fraction by canceling out the common factor of (x - 1) from both the numerator and the denominator. However, it is essential to remember that this cancellation is valid only if x ≠ 1, since division by zero is undefined. This condition is a critical aspect of the problem and must be explicitly stated. After canceling the common factor, we get (f/g)(x) = -(x - 1), provided that x ≠ 1. Distributing the negative sign, we obtain the simplified expression (f/g)(x) = -x + 1, with the condition that x ≠ 1. This simplification is a significant step, as it presents the quotient function in a more manageable form. The condition x ≠ 1 is crucial because it defines the domain of the resulting function. The original function (f/g)(x) was undefined at x = 1, and this restriction remains even after simplification. Therefore, the final expression for (f/g)(x) is -x + 1, with the explicit constraint that x ≠ 1. This detailed process highlights the importance of factoring, simplifying, and considering domain restrictions when working with quotients of functions. Understanding these steps is essential for accurately solving similar problems and for developing a strong foundation in mathematical analysis. The exclusion of x = 1 from the domain is a key element that distinguishes this problem and showcases the nuances of function operations.

Evaluating (f/g)(1)

Now that we have (f/g)(x) = -x + 1 for x ≠ 1, let's evaluate (f/g)(1). Substituting x = 1 into our simplified expression, we get:

(f/g)(1) = -1 + 1 = 0

However, we must remember the restriction that x ≠ 1. This means that our simplified expression is not valid at x = 1. Looking back at the original expression, (f/g)(x) = (-x^2 + 2x - 1) / (x - 1), we see that if we substitute x = 1 directly, we get:

(f/g)(1) = (-1^2 + 2(1) - 1) / (1 - 1) = 0 / 0

This is an indeterminate form, which means the function is undefined at x = 1. Therefore, (f/g)(1) is undefined. To evaluate the quotient function at a specific point, we must consider the domain of the function. In this case, we want to find (f/g)(1), but as we found earlier, (f/g)(x) = -x + 1 only when x ≠ 1. This condition is crucial because it means we cannot directly substitute x = 1 into the simplified expression. If we were to substitute x = 1 into -x + 1, we would get -1 + 1 = 0. However, this would be incorrect because it ignores the original function's domain. To correctly evaluate (f/g)(1), we need to go back to the original expression: (f/g)(x) = (-x^2 + 2x - 1) / (x - 1). Substituting x = 1 into this expression, we get: (f/g)(1) = (-1^2 + 2(1) - 1) / (1 - 1) = (0) / (0). This result, 0/0, is an indeterminate form. An indeterminate form indicates that the function is undefined at that point. In this context, it means that (f/g)(1) is undefined because we cannot divide by zero. The indeterminate form 0/0 is a key concept in calculus and analysis. It often signals the presence of a removable singularity or a more complex behavior of the function near that point. In this case, the singularity at x = 1 is removable in the simplified expression -x + 1, but it remains a restriction in the original function. Therefore, the correct answer for (f/g)(1) is that it is undefined, which is often represented by the symbol ∅. This detailed analysis emphasizes the importance of considering the domain of a function before evaluating it at a specific point. It also highlights the significance of indeterminate forms in identifying points where a function is not defined. Understanding these concepts is crucial for a solid foundation in calculus and mathematical analysis. The process of evaluating functions must always include a check for domain restrictions to avoid erroneous conclusions. The result of (f/g)(1) being undefined underscores the nuanced nature of function evaluation and the critical role of domain considerations.

Conclusion

In summary, for the functions f(x) = -x^2 + 2x - 1 and g(x) = x - 1, we found that (f/g)(x) = -x + 1, provided that x ≠ 1. Evaluating (f/g)(1), we determined that the function is undefined at this point. This exercise demonstrates the importance of simplifying expressions while paying close attention to the domain of the resulting function. In conclusion, this article has provided a comprehensive analysis of finding the quotient of two functions and evaluating it at a specific point. We began by determining (f/g)(x) for the functions f(x) = -x^2 + 2x - 1 and g(x) = x - 1. Through factoring and simplification, we found that (f/g)(x) = -x + 1, with the crucial condition that x ≠ 1. This condition arises from the fact that the original expression, (-x^2 + 2x - 1) / (x - 1), is undefined when x = 1 due to division by zero. The process of simplifying algebraic expressions while carefully considering domain restrictions is a fundamental skill in mathematics. It ensures that we maintain the integrity of the function and avoid making incorrect conclusions. Next, we evaluated (f/g)(1). Substituting x = 1 into the simplified expression -x + 1 would yield 0. However, we must remember the restriction that x ≠ 1. Going back to the original expression, we found that substituting x = 1 results in the indeterminate form 0/0, which means (f/g)(1) is undefined. This highlights a critical concept in function evaluation: the domain of the function must always be considered. Evaluating a function outside its domain leads to undefined results or incorrect conclusions. The indeterminate form 0/0 is a signal that the function's behavior at that point requires further investigation, often involving concepts from calculus such as limits. In this case, while the simplified expression -x + 1 is defined at x = 1, the original function (f/g)(x) is not. Therefore, the correct answer for (f/g)(1) is undefined, denoted by ∅. This exercise underscores the importance of a thorough understanding of function operations, factoring, simplification, and domain restrictions. It also illustrates the subtle but crucial distinctions that can arise when dealing with rational functions. The ability to navigate these complexities is essential for success in advanced mathematical studies. By carefully considering each step and paying attention to the nuances of the problem, we can arrive at accurate and meaningful solutions. The key takeaways from this analysis include the importance of factoring, simplifying expressions, considering domain restrictions, and understanding the implications of indeterminate forms. These concepts are fundamental to a strong foundation in mathematics and will be invaluable in tackling more complex problems in the future.