Question

the function $f(x) = \frac{1}{x^2 - 9}$ is a rational function. answer parts (a) - (i).

c. check for symmetry. does the graph have y - axis symmetry, symmetry about the origin, or no symmetry? choose the
a. the graph of f is symmetric about the y - axis.
b. the graph of f is symmetric about the origin.
c. the graph has no symmetry.

d. find the y - intercept or state that the function does not have a y - intercept. select the correct choice and, if necessary
a. the y - intercept is $y = \square$.
(simplify your answer. type an integer or a simplified fraction.)
b. the function has no y - intercept.

Explanation:

Step1: Recall y-intercept definition

The y-intercept of a function \( f(x) \) is the value of \( f(0) \), provided \( x = 0 \) is in the domain of the function.

Step2: Substitute \( x = 0 \) into \( f(x) \)

Given \( f(x)=\frac{1}{x^{2}-9} \), substitute \( x = 0 \):
\( f(0)=\frac{1}{0^{2}-9}=\frac{1}{-9}=-\frac{1}{9} \).
Since \( x = 0 \) is in the domain (denominator \( 0^{2}-9=-9
eq0 \)), the function has a y-intercept.

Answer:

A. The y-intercept is \( y = -\frac{1}{9} \)