Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Which function has a graph that does not have a vertical asymptote? A. \(f(x)=\frac{1}{x^{2}+2}\) B. \(f(x)=\frac{1}{x^{2}-2}\) C. \(f(x)=\frac{3}{x^{2}}\) D. \(f(x)=\frac{2 x+1}{x-8}\)

Step-by-Step Solution

Answer

Function A does not have a vertical asymptote.

Step 1: Understand Vertical Asymptotes

A vertical asymptote occurs where the denominator of a function is zero, causing the function to be undefined. To determine if a function has a vertical asymptote, set the denominator equal to zero and solve for the variable.

Step 2: Analyze Function A

Function A is given by \(f(x)=\frac{1}{x^{2}+2}\). The denominator, \(x^{2}+2\), cannot be zero since \(x^{2}+2\) is always positive for all real numbers \(x\). Thus, this function does not have a vertical asymptote.

Step 3: Analyze Function B

Function B is \(f(x)=\frac{1}{x^{2}-2}\). Set the denominator \(x^{2}-2=0\), solving gives \(x=\pm\sqrt{2}\). This function has vertical asymptotes at \(x=\pm\sqrt{2}\).

Step 4: Analyze Function C

Function C is \(f(x)=\frac{3}{x^{2}}\). Set the denominator \(x^{2}=0\), solving gives \(x=0\). This function has a vertical asymptote at \(x=0\).

Step 5: Analyze Function D

Function D is \(f(x)=\frac{2x+1}{x-8}\). Set the denominator \(x-8=0\), solving gives \(x=8\). This function has a vertical asymptote at \(x=8\).

Step 6: Conclusion

Function A, \(f(x)=\frac{1}{x^{2}+2}\), is the only function that does not have a vertical asymptote because its denominator is never zero for any real number \(x\).

Key Concepts

Rational Functions

Rational functions are an important category of functions in mathematics. They are expressed as a ratio of two polynomials, meaning that they can be written in the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). Essentially, rational functions resemble fractions but instead of numbers, they consist of polynomials.

Key features of rational functions include:

• They can have discontinuities, which occur where the denominator polynomial is zero.
• They often depict asymptotic behavior, especially vertical asymptotes where the function tends to infinity.
• They are continuous and smooth in intervals where the denominator is not zero.

Understanding rational functions is important because they model real-world phenomena in subjects like physics and economics, where they can represent relationships between variables with restrictions or limits.

Denominator Zero

The concept of the denominator being zero is crucial in analyzing rational functions. A denominator that is zero makes the function undefined at that particular point on the graph.

When evaluating a rational function, identify the values of \( x \) that set the denominator \( Q(x) \) to zero:

• This is done by solving the equation \( Q(x) = 0 \).
• The solutions to this equation indicate where the function is not defined.

For example, in the function \( f(x)=\frac{1}{x^2-2} \), the points \( x=\pm\sqrt{2} \) cause the denominator to be zero, making the function undefined there.

This concept links directly to vertical asymptotes, where the function approaches infinity or negative infinity near these values. It's crucial for determining the behavior and characteristics of the function graph.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it approaches a certain line or curve. In the context of rational functions, we often talk about vertical and horizontal asymptotes.

• Vertical Asymptotes: Occur when the denominator of a rational function approaches zero. As \( x \) reaches a value that makes \( Q(x) = 0 \), the function tends to \( \pm \infty \), causing a vertical asymptote.
• Horizontal Asymptotes: Describe the behavior as \( x \) goes to positive or negative infinity. These are determined by the leading coefficients and degrees of the polynomials in the numerator and denominator.

Vertical asymptotes are particularly interesting because they indicate points where the function "explodes" to infinity.

Understanding these helps in sketching the graph correctly and predicting how the function acts outside typical bounds, providing insights into both local and global behavior of rational functions.