Function Quotient Calculation For F(x) And G(x)

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In the realm of mathematical functions, the operation of dividing one function by another introduces a fascinating dimension of analysis. When we delve into the quotient of functions, we are essentially examining how the output of one function relates to the output of another for the same input. This exploration not only unveils the behavior of individual functions but also reveals the intricate interplay between them. In this comprehensive discussion, we will focus on the specific functions f(x) = -x² + 2x - 1 and g(x) = x - 1, with the objective of finding (f/g)(x) and (f/g)(1). This journey will encompass algebraic manipulations, domain considerations, and the crucial identification of any values where the quotient may be undefined. The study of function quotients is a cornerstone in understanding more complex mathematical models and real-world applications, making this analysis a valuable endeavor for both students and enthusiasts of mathematics. The process involves finding the quotient function (f/g)(x) by dividing the expression for f(x) by the expression for g(x). Subsequently, we will simplify the resulting expression, keeping a close watch for any potential domain restrictions arising from the denominator. Once we have established the simplified form of (f/g)(x), we will then proceed to evaluate it at x = 1, carefully noting if this value falls within the defined domain of the quotient function. If the denominator becomes zero at a particular x value, the quotient is undefined at that point, which is a key aspect to consider. Understanding these nuances is critical in accurately interpreting the behavior of function quotients.

Defining the Functions

Before we embark on the process of finding the quotient, let's clearly define the functions we'll be working with. We are given two functions: f(x) = -x² + 2x - 1 and g(x) = x - 1. The function f(x) is a quadratic function, characterized by its parabolic shape when graphed. The negative coefficient of the x² term indicates that the parabola opens downwards, and the other terms influence its position and intercepts. Quadratic functions are fundamental in mathematics and physics, modeling a variety of phenomena such as projectile motion and the shape of suspension cables. The function g(x) = x - 1 is a linear function, represented graphically by a straight line. Its slope is 1, indicating a 45-degree angle with the x-axis, and it intersects the y-axis at -1. Linear functions are widely used in simple models where the rate of change is constant. Recognizing the nature of these individual functions is essential as we proceed to combine them in the quotient operation. Understanding the properties of each function, such as their domains and ranges, is crucial for correctly interpreting the quotient function. For instance, the domain of both f(x) and g(x) is all real numbers, but this does not automatically mean that the domain of (f/g)(x) will also be all real numbers, due to the potential for the denominator, g(x), to be zero. Therefore, a careful consideration of the denominator is a key step in finding the quotient of functions and its domain. The interplay between the numerator and denominator determines the overall behavior of the quotient function, including its asymptotes and discontinuities.

Finding (f/g)(x)

Now, let's delve into the core of our task: finding (f/g)(x). By definition, (f/g)(x) = f(x) / g(x). Substituting the given functions, we have (f/g)(x) = (-x² + 2x - 1) / (x - 1). The next crucial step is to simplify this expression. The numerator, -x² + 2x - 1, is a quadratic expression that can be factored. Factoring the numerator is a key algebraic technique that often reveals common factors with the denominator, leading to simplification of the expression. In this case, -x² + 2x - 1 can be factored as -(x² - 2x + 1), which further simplifies to -(x - 1)². Thus, we can rewrite (f/g)(x) as -(x - 1)² / (x - 1). Now we observe a common factor of (x - 1) in both the numerator and the denominator. Canceling this common factor is permissible as long as x ≠ 1, as division by zero is undefined. After canceling the common factor, we obtain (f/g)(x) = -(x - 1), which is a simplified linear function. It is paramount to remember the restriction x ≠ 1, as it creates a “hole” in the graph of the function at x = 1. This restriction highlights the importance of considering the domain when simplifying rational expressions. The domain of (f/g)(x) is all real numbers except x = 1, which we will need to keep in mind when evaluating the function at specific points. This simplification process not only makes the function easier to evaluate but also provides deeper insights into its behavior.

Evaluating (f/g)(1)

Having found the simplified expression for (f/g)(x) as -(x - 1), we now turn our attention to evaluating this quotient function at x = 1. Substituting x = 1 into the simplified expression, we get (f/g)(1) = -(1 - 1) = -0 = 0. However, we must revisit the crucial restriction we identified earlier: x ≠ 1. This restriction stems from the original expression for (f/g)(x), where the denominator, g(x) = x - 1, becomes zero when x = 1. Division by zero is undefined in mathematics, making the original quotient function undefined at x = 1. While our simplified expression yields a value of 0 at x = 1, it is essential to recognize that this is a consequence of the simplification process and does not reflect the true behavior of the original quotient function. The original function has a removable discontinuity, often referred to as a “hole,” at x = 1. Therefore, the correct answer for (f/g)(1) is that it is undefined. We denote this by the symbol ∅. This example underscores a critical point in working with rational functions: simplification can sometimes mask domain restrictions that are present in the original function. It is always necessary to consider the original function when determining the domain and evaluating the function at specific points.

Conclusion

In this exploration, we successfully navigated the process of finding the quotient of two functions, f(x) = -x² + 2x - 1 and g(x) = x - 1, and evaluating it at a specific point. We determined that (f/g)(x) = -(x - 1) with the crucial restriction that x ≠ 1. The simplification process involved factoring and canceling common factors, which is a common technique in dealing with rational functions. However, we also learned a vital lesson about the importance of considering the original function's domain before and after simplification. When evaluating (f/g)(1), we found that substituting x = 1 into the simplified expression gives 0. However, recognizing that the original function is undefined at x = 1 due to division by zero, we correctly concluded that (f/g)(1) is undefined, denoted as ∅. This problem highlights the significance of not only performing algebraic manipulations accurately but also understanding the underlying concepts of function domains and the implications of division by zero. Function quotients are a fundamental concept in calculus and other advanced mathematical topics, and mastering these skills is essential for a deeper understanding of mathematical analysis. The process of finding and evaluating function quotients involves a blend of algebraic skill and conceptual understanding, making it a valuable exercise in mathematical thinking.