Question

follow the seven step strategy to graph the following rational function. f(x) = \frac{3x}{x^2 - 4} to graph the function, first determine the symmetry of the graph of f. choose the correct answer below. \bigcirc y-axis symmetry \bigcirc neither y-axis symmetry nor origin symmetry \bigcirc origin symmetry

Explanation:

To determine the symmetry of the rational function \( f(x)=\frac{3x}{x^{2}-4} \), we use the tests for y - axis symmetry and origin symmetry.

Step 1: Test for y - axis symmetry

A function \( y = f(x) \) is symmetric about the y - axis if \( f(-x)=f(x) \) for all \( x \) in the domain.

Calculate \( f(-x) \):
Substitute \( x \) with \( -x \) in the function \( f(x)=\frac{3x}{x^{2}-4} \).
\( f(-x)=\frac{3(-x)}{(-x)^{2}-4}=\frac{- 3x}{x^{2}-4} \)

Now, compare \( f(-x) \) with \( f(x) \). We know that \( f(x)=\frac{3x}{x^{2}-4} \), and \( f(-x)=\frac{-3x}{x^{2}-4}
eq f(x) \) (since \( \frac{-3x}{x^{2}-4}\) is the negative of \( \frac{3x}{x^{2}-4} \) when \( x
eq0 \) and \( x^{2}-4
eq0 \)). So, the function does not have y - axis symmetry.

Step 2: Test for origin symmetry

A function \( y = f(x) \) is symmetric about the origin if \( f(-x)=-f(x) \) for all \( x \) in the domain.

We already found that \( f(-x)=\frac{-3x}{x^{2}-4} \) and \( -f(x)=-\frac{3x}{x^{2}-4}=\frac{-3x}{x^{2}-4} \)

Since \( f(-x) = - f(x) \), the function is symmetric about the origin.

Answer:

origin symmetry