Questions: f(x)=(x+9)/(x^2-81) B. There is no x-intercept. Select the correct choice below and fill in any answer boxes within your choice. A. The y-intercept is (0,-1/9). (Type an ordered pair, using integers or fractions.) B. There is no y-intercept. Find the vertical asymptote(s). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

Solution Steps

To solve the given problem, we need to:

1. Determine the y-intercept by evaluating \( f(0) \).
2. Find the vertical asymptotes by identifying the values of \( x \) that make the denominator zero.

1. To find the y-intercept, substitute \( x = 0 \) into the function \( f(x) \).
2. To find the vertical asymptotes, solve the equation \( x^2 - 81 = 0 \) for \( x \).

To find the y-intercept of the function \( f(x) = \frac{x + 9}{x^2 - 81} \), we substitute \( x = 0 \):

\[ f(0) = \frac{0 + 9}{0^2 - 81} = \frac{9}{-81} = -\frac{1}{9} \]

Thus, the y-intercept is \( (0, -\frac{1}{9}) \).

To find the vertical asymptotes, we set the denominator equal to zero:

\[ x^2 - 81 = 0 \]

Factoring gives:

\[ (x - 9)(x + 9) = 0 \]

This results in the solutions:

\[ x = 9 \quad \text{and} \quad x = -9 \]

Thus, the vertical asymptotes are at \( x = 9 \) and \( x = -9 \).

Final Answer