Problem 92 - College Algebra

Question

Which one of the following is true? a. The function \(f(x)=\frac{1}{\sqrt{x-3}}\) is a rational function. b. The \(x\) -axis is a horizontal asymptote for the graph of $$f(x)=\frac{4 x-1}{x+3}.$$ c. The number of televisions that a company can produce per week after \(t\) weeks of production is given by $$N(t)=\frac{3000 t^{2}+30,000 t}{t^{2}+10 t+25}.$$ Using this model, the company will eventually be able to produce \(30,000\) televisions in a single week. d. None of the given statements is true.

Step-by-Step Solution

Answer

Out of the given statements, only statement d is true.

Step 1: Evaluate Statement A

A rational function is a function that can be expressed as the quotient of two polynomials. The given function \(f(x)=\frac{1}{\sqrt{x-3}}\) cannot be expressed in this way because of the square root in the denominator, making Statement A false.

Step 2: Evaluate Statement B

A horizontal asymptote for the graph of a function exists if and only if the function approaches a finite value as x approaches infinity or negative infinity. Calculating the limit as x approaches infinity for the function \(f(x)=\frac{4x-1}{x+3}\), it can be found that the function approaches 4, not 0 (which would indicate the x-axis as a horizontal asymptote). Therefore, Statement B is also false.

Step 3: Evaluate Statement C

The statement indicates that the company will eventually be able to produce 30,000 televisions in a week. To evaluate this, compute the limit as t approaches infinity for the function \(N(t)=\frac{3000t^{2}+30,000 t}{t^{2}+10 t+25}\). If the limit equals 30,000, the statement is true. In this case, the limit as t approaches infinity is 3,000, not 30,000, making Statement C false.

Step 4: Evaluate Statement D

Statement D says that none of the other statement are true. Statement A, B and C have been evaluated as false, thus making Statement D true.

Key Concepts

Rational Functions

Rational functions are a fundamental concept in algebra. A rational function is a fraction where both the numerator and the denominator are polynomials. For example, a function like \( f(x) = \frac{2x^2 + 3x + 1}{x - 1} \) is a rational function. Key characteristics include:

• The numerator and denominator must both be polynomials.
• The denominator cannot be zero as divisions by zero are undefined.

Rational functions may look different from simple fractions because they can include higher-degree polynomials.
In the given exercise, the function \(f(x)=\frac{1}{\sqrt{x-3}}\) is not a rational function because the denominator does not qualify as a polynomial due to the square root.

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as the input (often \(x\)) becomes very large or very small. They inform us about the end behavior of a function. Here’s how to determine the horizontal asymptote of a rational function:

• Compare the degrees of the polynomials in the numerator and the denominator.
• If the degrees are the same, the horizontal asymptote is the coefficient ratio of the highest terms.
• If the degree of the numerator is less than the denominator, the horizontal asymptote is \(y = 0\).
• If the degree of the numerator is greater, there’s no horizontal asymptote.

In the example \(f(x)=\frac{4x-1}{x+3}\), both the numerator and denominator have degree 1, so the horizontal asymptote is \(y = \frac{4}{1} = 4\).
This explains why the \(x\)-axis is not the horizontal asymptote.

Limits

Limits are used to describe the value that a function approaches as the input approaches some value. They are foundational to calculus and help in understanding the behavior of functions over the long term. For a function \(f(x)\), the limit as \(x\) approaches a particular value \(a\) is written as \(\lim_{x \to a} f(x)\).
If we evaluate the limit as \(x\) approaches infinity for the function \(f(x) = \frac{4x-1}{x+3}\), we get \(4\). This tells us the value the function approaches as \(x\) grows larger and is directly linked to determining horizontal asymptotes.
Calculating limits such as \(\lim_{t \to \infty} N(t) = 3000\) for the production function model \(N(t)=\frac{3000t^{2}+30,000 t}{t^{2}+10 t+25}\), helps us understand that production seems to stabilize at 3,000.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as the input grows very large or very small. This is crucial for understanding the long-term behavior of functions. Key aspects include:

• How close the function gets to the asymptote.
• The direction from which the function approaches the asymptote.
• Whether the function crosses the asymptote.

In this exercise, the function \(N(t)=\frac{3000t^{2}+30,000 t}{t^{2}+10 t+25}\) demonstrates asymptotic behavior by stabilizing around a particular value for large \(t\).
This behavior helps predict outcomes such as maximum production levels or long-run business performance. Here, the model shows an asymptotic value of 3,000 televisions, not 30,000, as \(t\) becomes very large.