Determine algebraically whether the function is even odd, or neither. f(x)=(-x^3)/(5x^2)-6

To determine whether the function \( f(x) = \frac{-x^3}{5x^2 - 6} \) is even, odd, or neither, we need to analyze the function's behavior under the transformation \( x \to -x \).<br /><br />1. **Even Function**: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \).<br />2. **Odd Function**: A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain of \( f \).<br /><br />Let's compute \( f(-x) \):<br /><br />\[ f(-x) = \frac{-(-x)^3}{5(-x)^2 - 6} \]<br /><br />Simplify the expression:<br /><br />\[ f(-x) = \frac{-(-x)^3}{5(-x)^2 - 6} = \frac{-(-x^3)}{5x^2 - 6} = \frac{x^3}{5x^2 - 6} \]<br /><br />Now compare \( f(-x) \) with \( f(x) \):<br /><br />\[ f(x) = \frac{-x^3}{5x^2 - 6} \]<br />\[ f(-x) = \frac{x^3}{5x^2 - 6} \]<br /><br />Clearly, \( f(-x) \neq f(x) \), so the function is not even.<br /><br />Next, check if \( f(-x) = -f(x) \):<br /><br />\[ -f(x) = -\left( \frac{-x^3}{5x^2 - 6} \right) = \frac{x^3}{5x^2 - 6} \]<br /><br />We see that \( f(-x) = -f(x) \). Therefore, the function \( f(x) = \frac{-x^3}{5x^2 - 6} \) is an odd function.<br /><br />So, the function is **odd**.