Questions: Find all asymptotes, x-intercepts, and y-intercepts for the graph of the rational function and sketch the graph of the function. f(x) = (5-x^2)/(x^2-36) Find all vertical asymptotes. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. A. The equation of the vertical asymptote is □ (Type an equation.) B. There are two vertical asymptotes. The leftmost asymptote is □ The rightmost asymptote is □ (Type equations.) C. There are no vertical asymptotes.

Solution Steps

To find the vertical asymptotes, we need to determine where the denominator of the rational function is equal to zero. The function is given by:

\[ f(x) = \frac{5 - x^2}{x^2 - 36} \]

The denominator is \( x^2 - 36 \). Setting it to zero:

\[ x^2 - 36 = 0 \]

Solving for \( x \):

\[ x^2 = 36 \] \[ x = \pm 6 \]

So, the vertical asymptotes are at \( x = -6 \) and \( x = 6 \).

To find the x-intercepts, we need to determine where the numerator of the rational function is equal to zero. The numerator is \( 5 - x^2 \). Setting it to zero:

\[ 5 - x^2 = 0 \]

Solving for \( x \):

\[ x^2 = 5 \] \[ x = \pm \sqrt{5} \]

So, the x-intercepts are at \( x = \sqrt{5} \) and \( x = -\sqrt{5} \).

To find the y-intercept, we need to evaluate the function at \( x = 0 \):

\[ f(0) = \frac{5 - 0^2}{0^2 - 36} = \frac{5}{-36} = -\frac{5}{36} \]

So, the y-intercept is at \( y = -\frac{5}{36} \).

Final Answer