Question

follow the seven step strategy to graph the following rational function.
$f(x) = -\frac{2}{x^2 - 9}$
...
to graph the function, first determine the symmetry of the graph of f. choose the correct answer below.
\bigcirc neither y-axis symmetry nor origin symmetry
\bigcirc y-axis symmetry
\bigcirc origin symmetry

Explanation:

Step1: Recall symmetry tests

For a function \( f(x) \), to check y - axis symmetry, we test if \( f(-x)=f(x) \). To check origin symmetry, we test if \( f(-x)=-f(x) \).

Step2: Compute \( f(-x) \)

Given \( f(x)=-\frac{2}{x^{2}-9} \), substitute \( x \) with \( -x \):
\( f(-x)=-\frac{2}{(-x)^{2}-9} \)
Since \( (-x)^{2}=x^{2} \), we have \( f(-x)=-\frac{2}{x^{2}-9} \)

Step3: Compare \( f(-x) \) with \( f(x) \) and \( -f(x) \)

We see that \( f(-x)=-\frac{2}{x^{2}-9}=f(x) \). So the function satisfies the condition for y - axis symmetry (\( f(-x) = f(x) \)). It does not satisfy origin symmetry because \( f(-x)
eq - f(x) \) (since \( -f(x)=\frac{2}{x^{2}-9}
eq f(-x) \)).

Answer:

B. y - axis symmetry