Problem 94 - College Algebra with Modeling and Visualization

Question

If the parking attendants can wait on 5 vehicles per minute, the average time \(T\) in mimutes spent waiting in line and paying the attendant becomes \(T(x)=-\frac{1}{x-5}\) (a) What is a reasonable domain for \(T ?\) (b) Graph \(y=T(x) .\) Include any vertical asymptotes. (c) Explain what happens to \(T(x)\) as \(x \rightarrow 5\)

Step-by-Step Solution

Answer

(a) Domain: \( x > 5 \). (b) Graph: L-shaped curve with a vertical asymptote at \( x = 5 \). (c) \( T(x) \rightarrow -\infty \) as \( x \rightarrow 5 \).

Step 1: Determine the Reasonable Domain for T(x)

The function given is \( T(x) = -\frac{1}{x-5} \). A function's domain includes all values of \( x \) where the function is defined. Here, the function is undefined when the denominator is zero, which occurs at \( x = 5 \). Thus, the reasonable domain excludes \( x = 5 \), so the domain is \( x eq 5 \). Moreover, since \( x \) represents the number of vehicles handled and it must be positive, the domain is \( x > 5 \).

Step 2: Graph the Function T(x)

To graph \( y = T(x) = -\frac{1}{x-5} \), consider the vertical asymptote and the general shape of the hyperbola. The vertical asymptote occurs at \( x = 5 \), where the function is undefined. As \( x \) increases from 5, \( T(x) \) is negative and approaches zero, while as \( x \) approaches from the right, the values of \( T(x) \) decay towards negative infinity. Similarly, for \( x < 5 \), the function would approach negative values from positive infinity. Draw these trends to form an L shaped graph, avoiding \( x = 5 \).

Step 3: Analyze the Behavior as x Approaches 5

As \( x \rightarrow 5^+ \) (values approaching 5 from the right), \( T(x) \rightarrow -\infty \). This represents an increasingly large negative wait time, indicating instability or an undefined physical situation as the capacity approaches the limit. Similarly, as \( x \rightarrow 5^- \), \( T(x) \) would approach infinitely large negative values, indicating a similar undefined state.

Key Concepts

functions and their domains

Understanding functions and their domains is a fundamental concept in college algebra. Essentially, a function is a mathematical rule that assigns a unique output for every valid input value. The domain of a function, then, includes all the possible input values (often called 'x-values') that the function can accept without experiencing undefined behavior, such as division by zero or taking the square root of a negative number.

In the original exercise, the function given is \( T(x) = -\frac{1}{x - 5} \). Since division by zero would make the function undefined, we must exclude any value of \( x \) that makes the denominator zero. Here, that means \( x = 5 \). Therefore, the function \( T(x) \) is undefined at \( x = 5 \), so this value can't be a part of the domain.

Moreover, because \( x \) represents the number of vehicles, it must be positive and greater logically. This implies that \( x \) must be greater than 5, thus forming the reasonable domain as \( x > 5 \). This allows you to correctly determine the set of values \( x \) can logically and mathematically have in this context.

graphing functions

Graphing a function offers a visual way to understand its behavior and characteristics. For the function \( T(x) = -\frac{1}{x - 5} \), the act of graphing involves acknowledging key features such as intercepts, symmetry, and asymptotic behavior.

First, identify the vertical asymptote. For \( T(x) \), the vertical asymptote is at \( x = 5 \), where the function becomes undefined. This asymptote indicates there is no crossing or touching of this vertical line in the graph.

• To the right of \( x = 5 \), as \( x \) increases, the values of \( T(x) \) move towards zero but remain negative, forming one branch of the hyperbola.
• To the left of \( x = 5 \), if \( x \) were theoretically less than 5, \( T(x) \) would start from negative infinity, moving downwards and creating the other branch.

Since the function behaves like a standard hyperbola, it visually appears as an "L" shape, with both branches approaching, but never touching, the horizontal and vertical asymptotes.

asymptotic behavior

Asymptotic behavior refers to how a function behaves as the input values approach certain points, often infinity or a specific number where the function is undefined. In the context of the given function \( T(x) = -\frac{1}{x-5} \), its asymptotic behavior is crucial for understanding its limitations and the effects on wait times.

As \( x \) approaches 5 from either direction, \( T(x) \) tends towards negative infinity. Specifically, when approaching from the right (\( x \to 5^+ \)), \( T(x) \) decreases without bound, showing that the wait time estimation becomes less defined and increasingly negative, which corresponds to a mathematical representation of an impossible scenario in real-world terms.

• Approaching from the left (\( x \to 5^- \)) similarly extends towards negative infinity, confirming the function is undefined precisely at \( x = 5 \).
• This asymptotic behavior signals an impossibility in the real-world context, indicating inherent constraints or a breakdown in the model at and near this point.

Therefore, by recognizing these asymptotic hints, one gains insight into both the theoretical and practical limitations of the model described by the function.