Problem 12 - A Graphical Approach to College Algebra

Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. State the domain of \(f .\) $$f(x)=\frac{2 x+6}{x-4}$$

Step-by-Step Solution

Answer

Vertical asymptote at \( x = 4 \), horizontal asymptote at \( y = 2 \), and domain is all real numbers except \( x = 4 \).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function is equal to zero, because the function is undefined at these points. For \( f(x) = \frac{2x + 6}{x - 4} \), set the denominator equal to zero: \[ x - 4 = 0 \]Solve for \( x \):\[ x = 4 \]Thus, there is a vertical asymptote at \( x = 4 \).

Step 2: Identify Horizontal Asymptotes

To find horizontal asymptotes, compare the degrees of the numerator and denominator. The function \( f(x) = \frac{2x + 6}{x - 4} \) has both a numerator and a denominator of degree 1. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. Here, the leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 1. Thus, the horizontal asymptote is:\[ y = \frac{2}{1} = 2 \]

Step 3: Identify Oblique Asymptotes

Oblique or slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. For \( f(x) = \frac{2x + 6}{x - 4} \), since the degrees of the numerator and the denominator are equal, there's no oblique asymptote.

Step 4: Determine the Domain of the Function

The domain of the function consists of all real numbers except those that make the denominator zero. From Step 1, we found the denominator is zero at \( x = 4 \). Hence, the domain of \( f(x) \) is all real numbers except \( x = 4 \), which is written as:\[ x eq 4 \]

Key Concepts

Understanding Vertical Asymptotes

Vertical asymptotes are lines where a function's value heads toward infinity, either positive or negative. They occur at points where the denominator of a rational function is zero, making the function undefined. To locate vertical asymptotes in a rational function like \( f(x) = \frac{2x + 6}{x - 4} \), we must determine when the denominator equals zero.

• Set the denominator equal to zero: \( x - 4 = 0 \)
• Solve for \( x \), resulting in \( x = 4 \)

Thus, the vertical asymptote is at \( x = 4 \). This means as \( x \) approaches 4, the function's values move to either positive or negative infinity. Vertical asymptotes directly affect the graph, causing it to shoot up or down sharply at these points.

Exploring Horizontal Asymptotes

Horizontal asymptotes provide information about the behavior of a function as \(x\) approaches infinity. They represent a horizontal line that the graph of the function approaches as \(x\) becomes very large or very small. In the function \( f(x) = \frac{2x + 6}{x - 4} \), the degrees of both the numerator and denominator are equal (both are degree 1).

• When degrees are equal, the horizontal asymptote is found by dividing the leading coefficients.
• The leading coefficient of the numerator is 2 and of the denominator is 1.

Therefore, the horizontal asymptote is \( y = \frac{2}{1} = 2 \). This suggests that as \(x\) approaches positive or negative infinity, the value of \(f(x)\) will get closer and closer to 2, never quite reaching it.

Defining the Domain of a Rational Function

The domain of a rational function includes all possible real number values that \(x\) can take. However, it excludes points where the function is undefined. For \( f(x) = \frac{2x + 6}{x - 4} \), the function is undefined when the denominator is zero, and we previously found this occurs at \( x = 4 \).To determine the domain:

• Note that the function is undefined wherever the denominator is zero.
• The solution to \( x - 4 = 0 \) is \( x = 4 \).

This means the domain of \(f\) is all real numbers except \(x = 4\). In interval notation, this is expressed as \((-\infty, 4) \cup (4, \infty)\). The domain helps us know where the function "lives" and where it might have interruptions or breaks in its graph.