Work each problem. Which function has a graph that does not have a vertical asymptote?
A. ƒ(x)=1/(x²+2)
B. ƒ(x)=1/(x²-2)
C. ƒ(x)=3/x²
D. ƒ(x)=(2x+1)/(x-8)

Verified step by step guidance

1. Recall that vertical asymptotes occur where the denominator of a rational function is equal to zero, causing the function to be undefined at those points.
2. For each function, identify the denominator and set it equal to zero to find potential vertical asymptotes.
3. For option A: The denominator is \(x^2 + 2\). Set \(x^2 + 2 = 0\) and solve for \(x\).
4. For option B: The denominator is \(x^2 - 2\). Set \(x^2 - 2 = 0\) and solve for \(x\).
5. For option C: The denominator is \(x^2\). Set \(x^2 = 0\) and solve for \(x\). For option D: The denominator is \(x - 8\). Set \(x - 8 = 0\) and solve for \(x\). Then determine which function's denominator has no real solutions, indicating no vertical asymptotes.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in the graph of a function where the function approaches infinity or negative infinity as the input approaches a specific value, typically where the denominator of a rational function is zero and the numerator is nonzero.

Domain of Rational Functions

The domain of a rational function includes all real numbers except where the denominator equals zero, since division by zero is undefined. Identifying these values helps determine where vertical asymptotes may exist.

Analyzing Quadratic Expressions in Denominators

Quadratic expressions in denominators can be factored or analyzed using the discriminant to find real roots. If the quadratic has no real roots, the denominator never equals zero, meaning no vertical asymptotes occur.