Problem 30 - College Algebra with Modeling and Visualization

Question

Find any horizontal or vertical asymptotes. $$ f(x)=\frac{x}{x^{3}-x} $$

Step-by-Step Solution

Answer

Horizontal: \( y = 0 \), Vertical: \( x = 1, x = -1 \).

Step 1: Identify Horizontal Asymptotes

Horizontal asymptotes are found by comparing the degrees of the polynomial in the numerator and the denominator. Here, the numerator has degree 1 and the denominator has degree 3.Since the degree of the numerator (1) is less than the degree of the denominator (3), there is a horizontal asymptote at:\[ y = 0 \]

Step 2: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero, provided that the numerator is not zero at the same points. For our function, set the denominator equal to zero:\[ x^3 - x = 0 \]Factor the equation:\[ x(x^2 - 1) = 0 \]Further factor:\[ x(x - 1)(x + 1) = 0 \]The solutions are \( x = 0 \), \( x = 1 \), and \( x = -1 \).Next, check the numerator at these points:- At \( x = 0 \), \( x = 0 \).- At \( x = 1 \), \( x = 1 \).- At \( x = -1 \), \( x = -1 \).Since the numerator is not zero at \( x = 1 \) and \( x = -1 \), \( x = 1 \) and \( x = -1 \) are vertical asymptotes.

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes are a way to understand the behavior of a function as the input (\(x \)) grows very large in either direction. Essentially, these asymptotes help us determine what value a function approaches when \(x \) becomes infinitely large or small.

To determine horizontal asymptotes, we compare the degrees of the polynomial in the numerator and the denominator.

• If the degree of the numerator is less than the denominator, the horizontal asymptote is at \(y = 0 \).
• If the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients.
• If the degree of the numerator is greater than that of the denominator, there is no horizontal asymptote.

In the given function, \( f(x) = \frac{x}{x^{3}-x} \), the degree of the numerator is 1, and the degree of the denominator is 3.

Since 1 is less than 3, the function has a horizontal asymptote at \( y = 0 \). This implies that as \(x \) tends towards positive or negative infinity, \(f(x)\) approaches 0.

Vertical Asymptotes

Vertical asymptotes occur at the values of \(x\) where a function is undefined, often due to division by zero. These asymptotes show the points where a function "blows up" or doesn't produce a usual value.

To find vertical asymptotes in the function \( f(x) = \frac{x}{x^{3} - x} \), set the denominator equal to zero and solve:1. Start with \(x^3 - x = 0\).2. Factor the expression: \(x(x^2 - 1) = 0\).3. Further factor \(x^2 - 1\) into \((x-1)(x+1)\), so the equation becomes \(x(x - 1)(x + 1) = 0\).This gives you the potential vertical asymptotes at \(x = 0\), \(x = 1\), and \(x = -1\).

To confirm these, ensure the numerator is not zero at these points:

• At \(x = 0\), the numerator is 0, so this is not a vertical asymptote.
• At \(x = 1\), the numerator is 1, thus \(x = 1\) is a vertical asymptote.
• At \(x = -1\), the numerator is -1, thus \(x = -1\) is a vertical asymptote.

Hence, the vertical asymptotes are located at \(x = 1\) and \(x = -1\).

Polynomial Degrees

Understanding polynomial degrees is essential to analyze the behavior of rational functions, particularly when determining asymptotes. The degree of a polynomial is the highest power of the variable present in the expression, indicating how many times the variable is multiplied by itself.

In a function like \(f(x) = \frac{x}{x^{3}-x}\), knowing the degree of both the numerator and the denominator helps identify asymptotic behavior.

• The numerator \(x\) is a first-degree polynomial since the highest power of \(x\) is 1.
• The denominator \(x^3 - x\) is a third-degree polynomial as the highest power of \(x\) is 3.

Polynomial degrees not only determine horizontal asymptotes but also influence our understanding of the graph's overall shape.

They allow us to gauge which part grows faster as \(x\) approaches infinity, thereby indicating the function's end behavior. Mastering the concept of polynomial degrees enables deeper insight into calculus and graphing rational functions.