Problem 468 - Precalculus

Question

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions. $$f(x)=\frac{3+x}{x^{3}-27}$$

Step-by-Step Solution

Answer

Domain: all real numbers except \(x = 3\). Vertical asymptote at \(x = 3\), horizontal asymptote at \(y = 0\).

Step 1: Identify the Domain

The domain of the function is all real values of "x" for which the function is defined. This function has a denominator of \(x^3 - 27\). Set the denominator equal to zero and solve for "x" to find where the function is undefined: \(x^3 - 27 = 0\). Solving this gives \(x^3 = 27\), so \(x = 3\). The domain of \(f(x)\) is all real numbers except \(x = 3\).

Step 2: Find the Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero and the numerator is non-zero. In this function, the denominator \(x^3 - 27 = 0\) at \(x = 3\). The numerator, \(3 + x\), is non-zero at \(x = 3\) (since \(3+3=6\)). Therefore, there is a vertical asymptote at \(x = 3\).

Step 3: Identify the Horizontal Asymptotes

Horizontal asymptotes are determined by the degrees of the polynomial in the numerator and the denominator. The degree of the numerator, \(3 + x\), is 1, and the degree of the denominator, \(x^3 - 27\), is 3. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial because it tells us all possible inputs (or "x" values) that the function can accept. For rational functions like \(f(x) = \frac{3+x}{x^3 - 27}\), the domain is all real numbers except where the denominator equals zero. This is because division by zero is undefined in mathematics.

To find where the function \(f(x)\) is undefined, we need to solve the equation \(x^3 - 27 = 0\). Solving this equation helps us find values of \(x\) that we need to exclude from the domain. When we set \(x^3 - 27 = 0\), we get \(x^3 = 27\) leading to \(x = 3\). Hence, the domain of the function \(f(x)\) is all real numbers except \(x = 3\). In set notation, the domain is \( \{ x \in \mathbb{R} ~|~ x eq 3 \} \).

Recipes for checking domains of rational functions:

• Identify the denominator.
• Set it equal to zero and solve for "x".
• The domain excludes solutions to the equation.

Vertical Asymptotes

Vertical asymptotes are essentially invisible vertical lines that a graph approaches but never touches. They usually occur where the function is undefined, typically when the denominator of a rational function equals zero. For \(f(x) = \frac{3+x}{x^3 - 27}\), we already found that the function is undefined at \(x = 3\). Thus, it suggests a vertical asymptote.

To confirm a vertical asymptote, check if the numerator is not zero at the excluded domain points. Since the numerator \(3 + x\) equals 6 at \(x = 3\), the vertical asymptote at \(x = 3\) is confirmed.

Steps to determine vertical asymptotes:

• Set the denominator equal to zero and solve for "x".
• Ensure the numerator is non-zero for these "x" values.
• If both conditions are met, a vertical asymptote exists at these "x" values.

Horizontal Asymptotes

Horizontal asymptotes indicate the behavior of a graph as \(x\) approaches infinity or negative infinity. For rational functions, these are determined by comparing the degrees of the numerator and the denominator's polynomials.

In the function \(f(x) = \frac{3+x}{x^3 - 27}\), the degree of the numerator is 1 (because of "\(x\)") and the degree of the denominator is 3 (because of "\(x^3\)"). When the degree of the numerator is less than the denominator, as is our case, the horizontal asymptote is automatically \(y = 0\).

Determining horizontal asymptotes:

• Compare the degrees of the numerator and the denominator.
• If the numerator's degree is less, \(y = 0\) is the asymptote.
• Makes the graph stabilize at \(y = 0\) as \(x\) becomes extremely large.