Problem 14 - College Algebra

Question

For the following exercises, find the domain, vertical asymptotes, and horizontes 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: Find the Domain

The domain of a function is the set of all possible input values (x-values) that the function can accept without leading to undefined expressions. For the function \( f(x) = \frac{3+x}{x^3 - 27} \), the expression in the denominator \( x^3 - 27 \) cannot be zero, since division by zero is undefined. Therefore, set the denominator equal to zero and solve for \( x \): \[ x^3 - 27 = 0 \]\[ x^3 = 27 \]\[ x = \sqrt[3]{27} = 3 \]Thus, the domain of \( f(x) \) is all real numbers except \( x = 3 \).

Step 2: Identify Vertical Asymptotes

A vertical asymptote occurs where the function tends towards infinity as the value of \( x \) approaches a certain point. This typically happens where the denominator of a rational function equals zero. From Step 1, we know the denominator is zero at \( x = 3 \). Thus, \( x = 3 \) is a vertical asymptote.

Step 3: Determine Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \( x \) approaches infinity or negative infinity. For rational functions of the form \( \frac{P(x)}{Q(x)} \), the horizontal asymptote is determined by the degrees of \( P(x) \) and \( Q(x) \). Here, the degree of the numerator \( 3+x \) is 1, and the degree of the denominator \( x^3-27 \) is 3. When the degree of the denominator is greater than that of the numerator, the horizontal asymptote is \( y = 0 \).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial because it identifies all possible x-values that can be plugged into the function without causing any mathematical issues such as division by zero. For the rational function \( f(x) = \frac{3+x}{x^3 - 27} \), attention should be paid to the denominator. When the denominator is zero, the function will be undefined, which restricts the domain.

Here's how you determine it:

• Set the denominator equal to zero: \( x^3 - 27 = 0 \).
• Solve for \( x \): \( x^3 = 27 \).
• Take the cube root: \( x = 3 \).

Therefore, the domain of \( f(x) \) is all real numbers except \( x = 3 \), written as \( (-\infty, 3) \cup (3, \infty) \). This approach helps ensure that we don't have division by zero, resulting in a clean and valid function.

Vertical Asymptotes

Vertical asymptotes are vertical lines where a function's value tends towards positive or negative infinity as it gets closer to these lines. For rational functions, vertical asymptotes typically occur where the denominator is zero and the numerator is not zero.

Following our previous analysis of the function \( f(x) = \frac{3+x}{x^3 - 27} \), we have:

• The denominator becomes zero at \( x = 3 \).
• Thus, \( x = 3 \) is identified as a vertical asymptote.

This means, as \( x \) approaches 3 from either side, the function \( f(x) \) tends towards infinity or negative infinity. Understanding vertical asymptotes helps in sketching the graph of a rational function, revealing where it might shoot up or down drastically.

Horizontal Asymptotes

Horizontal asymptotes represent the behavior of a function as \( x \) moves towards infinity or negative infinity. They give us a boundary line that the function's graph approaches but may never actually reach.

For a rational function of the form \( \frac{P(x)}{Q(x)} \), identifying the horizontal asymptote depends on the degrees of the numerator and the denominator:

• If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is \( y = 0 \).
• If the degree of the numerator and denominator are equal, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of the numerator and denominator respectively.

In our function \( f(x) = \frac{3+x}{x^3 - 27} \), the degree of the numerator is 1, and the degree of the denominator is 3. Hence, the horizontal asymptote is \( y = 0 \). This implies, as \( x \) heads to either positive or negative infinity, \( f(x) \) steadily approaches zero, aiding in understanding the end behavior of the function graph.