Problem 66 - Algebra for College Students

Question

In each case find a rational finction whose graph has the required asymptotes. Answers may vary. The graph has the line \(y=2\) as a horizontal asymptote and the line \(x=1\) as a vertical asymptote.

Step-by-Step Solution

Answer

The required function is \( f(x) = \frac{2x}{x-1} \).

Step 1 - Understand Asymptotes

Horizontal asymptote at y=2 means as x approaches infinity, the function approaches 2. Vertical asymptote at x=1 means the function is undefined at x=1.

Step 2 - Start with a Basic Rational Function

Consider a rational function of the form \( f(x) = \frac{N(x)}{D(x)} \), where N(x) and D(x) are polynomials.

Step 3 - Ensure Function Meets Vertical Asymptote

To have a vertical asymptote at x=1, the denominator should be zero at x=1. Choose D(x) = x - 1.

Step 4 - Ensure Function Meets Horizontal Asymptote

To have a horizontal asymptote at y=2, the degree of the numerator and denominator should be the same, and the leading coefficients' ratio should be 2. Choose N(x) = 2x.

Step 5 - Combine to Form the Function

Thus, the required rational function is \( f(x) = \frac{2x}{x-1} \).

Key Concepts

asymptotes

Asymptotes are lines that a graph approaches but never touches. They provide critical insights into the behavior of a rational function as input values become very large or very small.
There are mainly two types of asymptotes relevant to rational functions: horizontal and vertical.
Understanding how to identify and utilize these can help grasp the function's overall shape and behavior.

horizontal asymptotes

Horizontal asymptotes tell us about the behavior of a function as the input values (x) grow extremely large or extremely small. In the given exercise, the horizontal asymptote is at \( y = 2 \). This means that as \( x \) approaches infinity (positive or negative), the function value (\( y \)) approaches 2.

To find the horizontal asymptote for a rational function \( f(x) = \frac{N(x)}{D(x)} \), we look at the degrees of the polynomials in the numerator and the denominator.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
  
• If the degree of the numerator and the degree of the denominator are the same, the horizontal asymptote is the ratio of the leading coefficients.

In the exercise, the numerator and denominator are both linear (degree 1), and the leading coefficients are 2 and 1, respectively. Hence, the horizontal asymptote is \( y = 2 \).

vertical asymptotes

Vertical asymptotes are lines that indicate where the function becomes undefined because the denominator is zero. They show up as the inverse function is shooting off to positive or negative infinity.
In the given exercise, the vertical asymptote is at \( x = 1 \). This means that the function is undefined at \( x = 1 \) and as \( x \) approaches 1 from either side, \( f(x) \) approaches either positive or negative infinity.
To identify a vertical asymptote for a rational function \( f(x) = \frac{N(x)}{D(x)} \), find the value of \( x \) that makes \( D(x) = 0 \).
In our exercise, we chose the denominator \( D(x) = x - 1 \), making the function undefined at \( x = 1 \). This choice gives us the vertical asymptote at \( x = 1 \).
Understanding vertical asymptotes can help you avoid undefined values and better graph the overall function.