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 use the following definitions:<br /><br />1. A function is **even** if \( f(-x) = f(x) \) for all \( x \).<br />2. A function is **odd** if \( f(-x) = -f(x) \) for all \( x \).<br />3. If neither condition is satisfied, the function is **neither even nor odd**.<br /><br />---<br /><br />### Step 1: Compute \( f(-x) \)<br />Substitute \( -x \) into the function \( f(x) \):<br />\[<br />f(-x) = \frac{-( -x )^3}{5(-x)^2 - 6}.<br />\]<br /><br />Simplify the numerator and denominator:<br />- The numerator becomes:<br />\[<br />-( -x )^3 = -(-x^3) = x^3.<br />\]<br />- The denominator becomes:<br />\[<br />5(-x)^2 - 6 = 5x^2 - 6.<br />\]<br /><br />Thus:<br />\[<br />f(-x) = \frac{x^3}{5x^2 - 6}.<br />\]<br /><br />---<br /><br />### Step 2: Compare \( f(-x) \) with \( f(x) \)<br />The original function is:<br />\[<br />f(x) = \frac{-x^3}{5x^2 - 6}.<br />\]<br /><br />From Step 1, we found:<br />\[<br />f(-x) = \frac{x^3}{5x^2 - 6}.<br />\]<br /><br />Clearly, \( f(-x) \neq f(x) \), so the function is **not even**.<br /><br />Now 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 /><br />We see that \( f(-x) = -f(x) \). Therefore, the function is **odd**.<br /><br />---<br /><br />### Final Answer:<br />The function \( f(x) = \frac{-x^3}{5x^2 - 6} \) is **odd**.