Problem 18 - College Algebra

Question

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions. $$ f(x)=\frac{x-4}{x-6} $$

Step-by-Step Solution

Answer

Domain: \( x \neq 6 \); Vertical asymptote: \( x=6 \); Horizontal asymptote: \( y=1 \).

Step 1: Find the Domain

The domain of a rational function is all real numbers except where the denominator is zero. For the function \( f(x) = \frac{x-4}{x-6} \), set the denominator equal to zero: \( x-6=0 \). Solve for \( x \), giving \( x=6 \). Therefore, the domain is all real numbers except \( x=6 \): \( x \in \mathbb{R} \setminus \{6\} \).

Step 2: Find the Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero, provided the numerator is not zero at that point. From Step 1, we found \( x=6 \) makes the denominator zero. Since the numerator at \( x=6 \) (that is, \( 6-4=2 \)) is not zero, there is a vertical asymptote at \( x=6 \).

Step 3: Find the Horizontal Asymptotes

For rational functions, if the degree of the polynomial in the numerator is the same as in the denominator, the horizontal asymptote is the ratio of the leading coefficients. In \( f(x) = \frac{x-4}{x-6} \), both the numerator and denominator are first degree polynomials with leading coefficient 1. Thus, the horizontal asymptote is \( y = \frac{1}{1} = 1 \).

Key Concepts

Domain

In the world of rational functions, the domain refers to all possible input values that allow the function to operate correctly. Specifically, it includes all real numbers except those that make the denominator zero, as division by zero is undefined.

Let’s break down the reasoning with an example: Take the rational function \( f(x) = \frac{x-4}{x-6} \). Here, the denominator is \( x-6 \). To find values that are not allowed in the domain, we set the denominator to zero: \( x-6 = 0 \). Solving for \( x \), we get \( x = 6 \). This means \( x = 6 \) would make the denominator zero, causing the function to be undefined at this point.

• Therefore, the domain of \( f(x) \) is all real numbers except \( x = 6 \).
• This is expressed as \( x \in \mathbb{R} \setminus \{6\} \).

Keep in mind, identifying the domain is a crucial first step in understanding how rational functions behave.

Vertical Asymptotes

Vertical asymptotes are an intriguing feature of rational functions. They occur at points where the function approaches infinity as it nears specific \( x \)-values. These generally form where the denominator is zero, but unlike the domain, we also ensure that the numerator isn't zero at the same point.

For \( f(x) = \frac{x-4}{x-6} \), we've determined the denominator is zero at \( x = 6 \). To verify the presence of a vertical asymptote here, check the numerator at \( x = 6 \):

• The numerator \( x-4 \) at \( x=6 \) evaluates to \( 6-4=2 \).
• Since the numerator is not zero, a vertical asymptote exists at \( x = 6 \).

This results in the function skyrocketing towards infinity or plummeting towards negative infinity as \( x \) approaches 6, creating a vertical line that the graph never actually touches or crosses.

Horizontal Asymptotes

Horizontal asymptotes help understand the end behavior of a rational function as \( x \) approaches positive or negative infinity. For rational functions, how the degrees of the numerator and denominator compare determines the nature of any horizontal asymptotes.

Consider \( f(x) = \frac{x-4}{x-6} \), where both the numerator and denominator polynomials are of the first degree (degree 1). When their degrees are equal, the horizontal asymptote is the ratio of their leading coefficients.

• In this function, both leading coefficients are 1.
• Therefore, the horizontal asymptote is found at \( y = \frac{1}{1} = 1 \).

This simple ratio gives a line that the graph approaches, but does not necessarily intersect, as \( x \) moves towards the extremes of the number line. Thus, horizontal asymptotes describe the flattening behavior of a function's graph at infinity.