Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Find the domain of the function and identify any horizontal and vertical asymptotes. $$f(x)=\frac{x}{x-2}$$

Step-by-Step Solution

Answer

The domain of the function is all real numbers except when x = 2. The function has a vertical asymptote at x = 2 and a horizontal asymptote at y = 1.

Step 1: Determine the Domain

First, one needs to find out the values of \(x\) for which the function is defined. The function will be undefined when the denominator is zero, because you cannot divide by zero. So to find the domain, set the denominator equal to 0 and solve for \(x\), i.e., \(x-2 = 0\). This gives \(x=2\). Therefore the domain of the function is all real numbers except \(x=2\). Mathematically, this can be expressed as \(x \neq 2\).

Step 2: Identify the Vertical Asymptote

As mentioned, a vertical asymptote happens at the value of \(x\) that makes the denominator zero. In this case, it is \(x=2\). This means there is a vertical asymptote at \(x=2\)

Step 3: Identify the Horizontal Asymptote

Horizontal asymptotes are determined by looking at the degrees of the highest terms of the numerator and denominator. In this case, the degree of both the numerator and the denominator are equal. When this is the case, the horizontal asymptote is given by the ratio of the leading coefficients, which in this case is \(\frac{1}{1}\) or \(y=1\)

Key Concepts

Domain of a Function

The domain of a function refers to the complete set of values for which the function is defined. When dealing with rational functions, like \( f(x) = \frac{x}{x-2} \), it's crucial to identify values of \( x \) that make the function undefined. This often occurs when the denominator is zero, since division by zero is undefined.

In the given function, setting the denominator \( x-2 \) equal to zero, we find that \( x = 2 \) is the problematic value. Thus, the domain of \( f(x) \) can be described as all real numbers except \( x = 2 \).

When finding domains:

• Identify any values that cause the denominator to be zero.
• Exclude these values from the domain.
• Express the domain in set notation for clarity, for example, \( x eq 2 \).

Ensuring that one understands which values to exclude helps in evaluating the function accurately and prevents undefined operations.

Vertical Asymptotes

Vertical asymptotes represent the values of \( x \) where the function grows infinitely positive or negative as it approaches these values. Rational functions may have vertical asymptotes at points where their denominators are zero.

In our example, \( f(x) = \frac{x}{x-2} \), we found a vertical asymptote by solving \( x-2 = 0 \), giving \( x = 2 \). Therefore, there is a vertical asymptote at \( x=2 \). At this point, the function is undefined and draws near to infinity as \( x \) approaches 2 from either direction.


Understanding vertical asymptotes involves:

• Locating values where the denominator equals zero.
• Recognizing the function's behavior as it nears those values.
• Remembering that the function does not intersect vertical asymptotes.

This intrinsic property of asymptotes provides crucial insight into the graph's overall shape and behavior near those specific \( x \)-values.

Horizontal Asymptotes

Horizontal asymptotes give insight into how a function behaves as \( x \) approaches infinity or negative infinity. They indicate the value that the function approaches, but never actually meets, as \( x \) becomes very large or very small.

For a rational function, comparing the degrees of the polynomials in the numerator and denominator can help determine horizontal asymptotes. If the degrees are equal, like in \( f(x) = \frac{x}{x-2} \), the horizontal asymptote is the ratio of the leading coefficients. Here, both the numerator and denominator have a degree of 1, and their leading coefficients' ratio is \( \frac{1}{1} = 1 \). Thus, the horizontal asymptote is \( y = 1 \).

Key points about horizontal asymptotes include:

• The line \( y = 1 \) indicates the function's end behavior.
• A function can intersect its horizontal asymptote but will approach it as \( x \) moves towards infinity.
• Understanding horizontal asymptotes is essential for predicting the function's long-term behavior.

These guidelines help you visualize the tendencies of a function as its values expand."