Finding The Quotient Of Functions F(x) = 1/x And G(x) = X - 5

In the realm of mathematical functions, understanding how different functions interact and combine is crucial. One such interaction is the quotient of two functions, denoted as (f/g)(x), where we divide one function, f(x), by another, g(x). This operation unveils new insights into the behavior and properties of the resulting function. In this article, we delve into the concept of the quotient of functions, specifically focusing on the functions f(x) = 1/x and g(x) = x - 5. We will explore the process of finding (f/g)(x), discuss its domain, and analyze its characteristics. This exploration will not only enhance our understanding of function operations but also provide a foundation for more advanced mathematical concepts.

Before we dive into the quotient, it's essential to have a clear understanding of the individual functions involved. Let's start with f(x) = 1/x. This is a reciprocal function, where the output is the inverse of the input. As x approaches infinity, f(x) approaches zero, and as x approaches zero, f(x) approaches infinity. This function has a vertical asymptote at x = 0, indicating that the function is undefined at this point. The graph of f(x) = 1/x is a hyperbola, with two branches extending towards the asymptotes.

Next, consider g(x) = x - 5. This is a linear function with a slope of 1 and a y-intercept of -5. The graph of g(x) is a straight line that increases as x increases. Linear functions are fundamental in mathematics and have numerous applications in modeling real-world scenarios.

Understanding the behavior of these individual functions is crucial for comprehending their quotient. The characteristics of f(x) and g(x) will influence the domain, range, and overall behavior of (f/g)(x).

Now, let's move on to the main objective: finding the quotient function (f/g)(x). According to the definition of the quotient of functions, (f/g)(x) = f(x) / g(x), provided that g(x) ≠ 0. In our case, f(x) = 1/x and g(x) = x - 5. Therefore, (f/g)(x) = (1/x) / (x - 5).

To simplify this expression, we can rewrite the division as multiplication by the reciprocal of the denominator. This gives us (f/g)(x) = (1/x) * (1 / (x - 5)). Multiplying the numerators and denominators, we get (f/g)(x) = 1 / (x(x - 5)).

This resulting function, (f/g)(x) = 1 / (x(x - 5)), is a rational function. Rational functions are functions that can be expressed as the ratio of two polynomials. Understanding the process of finding the quotient function is essential for working with more complex function operations and applications.

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the quotient function (f/g)(x), we need to consider the domains of both f(x) and g(x), as well as any additional restrictions imposed by the division operation.

First, let's consider the domain of f(x) = 1/x. As we discussed earlier, f(x) is undefined when x = 0, as division by zero is not allowed. Therefore, the domain of f(x) is all real numbers except 0, which can be written as (-∞, 0) U (0, ∞).

Next, let's consider the domain of g(x) = x - 5. Since g(x) is a linear function, it is defined for all real numbers. Therefore, the domain of g(x) is (-∞, ∞).

Now, let's consider the quotient function (f/g)(x) = 1 / (x(x - 5)). In addition to the restrictions imposed by f(x), we also need to ensure that the denominator of (f/g)(x) is not equal to zero. This means that x(x - 5) ≠ 0. Solving this inequality, we find that x ≠ 0 and x ≠ 5.

Combining these restrictions, we find that the domain of (f/g)(x) is all real numbers except 0 and 5. This can be written as (-∞, 0) U (0, 5) U (5, ∞). Understanding the domain of a function is crucial for interpreting its behavior and applications.

Now that we have found the quotient function (f/g)(x) = 1 / (x(x - 5)) and determined its domain, let's analyze its characteristics. This analysis will provide insights into the behavior of the function and its relationship to the original functions, f(x) and g(x).

Asymptotes: One of the key characteristics of rational functions is the presence of asymptotes. Asymptotes are lines that the function approaches but never touches. We have already identified that (f/g)(x) has vertical asymptotes at x = 0 and x = 5, as these are the values that make the denominator equal to zero.

To determine if there are any horizontal asymptotes, we need to examine the behavior of the function as x approaches infinity and negative infinity. As x becomes very large (either positive or negative), the term x(x - 5) in the denominator also becomes very large. Therefore, the value of (f/g)(x) = 1 / (x(x - 5)) approaches zero. This indicates that there is a horizontal asymptote at y = 0.

Intercepts: Another important characteristic of a function is its intercepts, which are the points where the graph of the function crosses the x-axis and y-axis.

To find the x-intercepts, we need to solve the equation (f/g)(x) = 0. In our case, 1 / (x(x - 5)) = 0. However, a fraction can only be equal to zero if its numerator is equal to zero. Since the numerator of (f/g)(x) is 1, which is never zero, there are no x-intercepts.

To find the y-intercept, we need to evaluate (f/g)(0). However, we know that x = 0 is a vertical asymptote, so (f/g)(0) is undefined. Therefore, there is no y-intercept.

Behavior: To fully understand the behavior of (f/g)(x), we can analyze its sign in different intervals determined by the vertical asymptotes. The vertical asymptotes divide the domain into intervals (-∞, 0), (0, 5), and (5, ∞).

• In the interval (-∞, 0), both x and (x - 5) are negative, so their product x(x - 5) is positive. Therefore, (f/g)(x) = 1 / (x(x - 5)) is positive in this interval.
• In the interval (0, 5), x is positive and (x - 5) is negative, so their product x(x - 5) is negative. Therefore, (f/g)(x) = 1 / (x(x - 5)) is negative in this interval.
• In the interval (5, ∞), both x and (x - 5) are positive, so their product x(x - 5) is positive. Therefore, (f/g)(x) = 1 / (x(x - 5)) is positive in this interval.

This analysis, combined with the information about asymptotes and intercepts, provides a comprehensive understanding of the behavior of the quotient function (f/g)(x). Understanding the characteristics of functions is crucial for applications in various fields, including physics, engineering, and economics.

In this article, we explored the concept of the quotient of functions, focusing on the functions f(x) = 1/x and g(x) = x - 5. We successfully found the quotient function (f/g)(x) = 1 / (x(x - 5)), determined its domain as (-∞, 0) U (0, 5) U (5, ∞), and analyzed its characteristics, including asymptotes, intercepts, and behavior in different intervals. This exploration provided valuable insights into function operations and the properties of rational functions. The ability to find and analyze the quotient of functions is a fundamental skill in mathematics and has wide-ranging applications in various fields. By understanding these concepts, we can better interpret and model real-world phenomena using mathematical tools. The process of combining functions, such as finding their quotient, allows us to create more complex and nuanced models, enhancing our ability to solve problems and make predictions.