Problem 13 - Precalculus with Calculus Previews

Question

Find the vertical and horizontal asymptotes for the graph of the given rational function. Find \(x\) - and \(y\) -intercepts of the graph. Sketch the graph of \(f\). $$ f(x)=\frac{1}{x^{3}} $$

Step-by-Step Solution

Answer

Vertical asymptote: \( x = 0 \); Horizontal asymptote: \( y = 0 \); No intercepts.

Step 1: Identify Vertical Asymptotes

Vertical asymptotes are found where the denominator of the rational function equals zero. For the function \( f(x) = \frac{1}{x^3} \), the denominator is \( x^3 \). Setting the denominator equal to zero gives \( x^3 = 0 \). Solving this gives \( x = 0 \). Thus, there is a vertical asymptote at \( x = 0 \).

Step 2: Identify Horizontal Asymptotes

Horizontal asymptotes for a rational function of the form \( \frac{p(x)}{q(x)} \) can be found by comparing the degree of \( p(x) \) and \( q(x) \). Here, \( p(x) = 1 \) with degree 0, and \( q(x) = x^3 \) with degree 3. Since the degree of the denominator is greater than that of the numerator (\( 0 < 3 \)), the horizontal asymptote is \( y = 0 \).

Step 3: Find x-intercepts

The \( x \)-intercepts of a rational function occur when the numerator equals zero, as long as the zero is not also in the denominator. Since the numerator is \( 1 \) and not zero for any \( x \), there are no \( x \)-intercepts for \( f(x) = \frac{1}{x^3} \).

Step 4: Find y-intercepts

The \( y \)-intercept of a function is found by evaluating \( f(0) \). Since there's a vertical asymptote at \( x = 0 \), \( f(0) \) is undefined. Thus, there is no \( y \)-intercept for \( f(x) = \frac{1}{x^3} \).

Step 5: Sketch the Graph

To sketch the graph, note the vertical asymptote at \( x = 0 \) and the horizontal asymptote at \( y = 0 \). For \( x > 0 \), \( x^3 \) is positive, so \( f(x) \) is positive and decreases towards zero as \( x \) increases. For \( x < 0 \), \( x^3 \) is negative, so \( f(x) \) is negative and increases towards zero (from below) as \( x \) decreases. There are no intercepts, so the graph approaches the asymptotes without crossing the axes.

Key Concepts

Rational Functions

Rational functions are a type of function expressed as the quotient of two polynomials. In mathematical terms, a rational function is written as \( f(x) = \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials, and \( q(x) eq 0 \).
These functions are interesting because they can describe a wide range of behaviors depending on the degrees of the polynomials involved.

• If the numerator's degree is greater than the denominator's, the graph may have oblique asymptotes.
• If they have the same degree, a horizontal asymptote determined by the leading coefficients occurs.
• If the denominator's degree is larger, the horizontal asymptote is often the x-axis (\( y = 0 \)).

In our example, \( f(x) = \frac{1}{x^3} \), this is a simple rational function where the numerator is a constant (degree 0) and the denominator is a monomial of degree 3.

Vertical Asymptotes

A vertical asymptote is a vertical line that the graph of a function approaches but never touches or crosses. This occurs where the function is undefined, particularly when the denominator of a rational function equals zero.
To find the vertical asymptotes, set the denominator equal to zero and solve for \( x \).
In the function \( f(x) = \frac{1}{x^3} \), the denominator is \( x^3 \), which becomes zero when \( x = 0 \).
Therefore, there is a vertical asymptote at \( x = 0 \).
Vertical asymptotes often signify limits in behavior as the graph approaches infinities as it gets closer to these \( x \)-values from the sides.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a graph as \( x \) approaches positive or negative infinity. It represents a constant value that the function approaches but does not necessarily ever reach.
For rational functions, horizontal asymptotes are identified by comparing the degrees of the numerator and the denominator.

• If the degree of the denominator exceeds that of the numerator, the horizontal asymptote is \( y = 0 \).
• If the degrees are equal, the horizontal asymptote corresponds to the ratio of leading coefficients.
• If the numerator's degree is higher, no horizontal asymptote exists, but sometimes an oblique one does.

For our function \( f(x) = \frac{1}{x^3} \), the denominator has a higher degree than the numerator, resulting in a horizontal asymptote at \( y = 0 \).

Intercepts

Intercepts are the points where the graph of a function crosses the axes. In the context of rational functions, finding these involves a few straightforward steps.

• x-intercepts: Occur where the numerator of the function equals zero, providing it's not canceled by the denominator. However, since the numerator in \( f(x) = \frac{1}{x^3} \) is 1, which is never zero, this function has no \( x \)-intercepts.
• y-intercepts: Found by evaluating the function at \( x = 0 \). If the function value exists, this point is the \( y \)-intercept. In this case, since there is a vertical asymptote at \( x = 0 \), \( f(x) \) is undefined, meaning there is no \( y \)-intercept.

Understanding intercepts is crucial in sketching the graph of a function, along with asymptotes, to comprehend its overall shape and direction.
Thus, with no intercepts, the graph of \( f(x) = \frac{1}{x^3} \) will simply approach its asymptotes without intersecting the axes.