Work each problem. Which function has a graph that does not have a vertical asymptote?A. ƒ(x)=1/(x^2+2) B. ƒ(x)=1/(x^2-2) C. ƒ(x)=3/x^2 D. ƒ(x)=(2x+1)/(x-8)

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1. Identify the condition for a vertical asymptote: A vertical asymptote occurs where the denominator of a rational function is equal to zero, and the numerator is not zero at that point.
2. Analyze option A: \( f(x) = \frac{1}{x^2 + 2} \). The denominator \( x^2 + 2 \) is never zero for any real number \( x \) because \( x^2 \) is always non-negative and adding 2 makes it positive.
3. Analyze option B: \( f(x) = \frac{1}{x^2 - 2} \). Set the denominator equal to zero: \( x^2 - 2 = 0 \). Solve for \( x \) to find potential vertical asymptotes.
4. Analyze option C: \( f(x) = \frac{3}{x^2} \). Set the denominator equal to zero: \( x^2 = 0 \). Solve for \( x \) to find potential vertical asymptotes.
5. Analyze option D: \( f(x) = \frac{2x+1}{x-8} \). Set the denominator equal to zero: \( x - 8 = 0 \). Solve for \( x \) to find potential vertical asymptotes.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in rational functions when the denominator approaches zero while the numerator does not. This results in the function tending towards infinity or negative infinity. To identify vertical asymptotes, one must find the values of x that make the denominator zero, as these points indicate where the function is undefined.

Rational Functions

A rational function is a function represented by the ratio of two polynomials. The general form is ƒ(x) = P(x)/Q(x), where P and Q are polynomials. Understanding the behavior of rational functions, particularly how their graphs behave near the zeros of the denominator, is crucial for analyzing their asymptotic behavior.

Behavior of Functions at Infinity

The behavior of functions at infinity helps determine how they behave as x approaches very large or very small values. For rational functions, this involves analyzing the degrees of the numerator and denominator. If the degree of the denominator is greater than that of the numerator, the function approaches zero, which can indicate the absence of vertical asymptotes.