Determine algebraically whether the given function is even, odd, or neither. \( h(x)=\frac{3 x^{3}}{8 x^{2}+3} \) Neither Even Odd 1
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To determine if the function \( h(x) = \frac{3 x^{3}}{8 x^{2}+3} \) is even, odd, or neither, we need to evaluate \( h(-x) \) and compare it to \( h(x) \) and \( -h(x) \). First, calculating \( h(-x) \): \[ h(-x) = \frac{3 (-x)^{3}}{8 (-x)^{2} + 3} = \frac{-3 x^{3}}{8 x^{2} + 3} \] Next, we analyze: - \( h(-x) \) is not equal to \( h(x) \) (it doesn’t match the original). - \( h(-x) = -\frac{3 x^{3}}{8 x^{2} + 3} = -h(x) \). Since \( h(-x) = -h(x) \), the function is odd! So, the correct conclusion is that the function is **Odd**.