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

Question

UpStudy AI · Accepted Answer

**Step 1. Substitute $$-x$$ into the function.**

We start with the function:
$$
h(x)=\frac{3x^3}{8x^2+3}
$$
Substitute $$-x$$ for $$x$$:
$$
h(-x)=\frac{3(-x)^3}{8(-x)^2+3}
$$

**Step 2. Simplify the numerator and denominator.**

Recall that $$(-x)^3=-x^3$$ and $$(-x)^2=x^2$$. Thus,
$$
h(-x)=\frac{3(-x^3)}{8x^2+3}=\frac{-3x^3}{8x^2+3}
$$

**Step 3. Compare $$h(-x)$$ with $$h(x)$$.**

We already have:
$$
h(x)=\frac{3x^3}{8x^2+3}
$$
and
$$
h(-x)=\frac{-3x^3}{8x^2+3} = -\frac{3x^3}{8x^2+3} = -h(x)
$$

**Step 4. Conclusion.**

Since
$$
h(-x) = -h(x)
$$
the function $$h(x)$$ is $$\textbf{odd}$$.
The function $$ h(x) = \frac{3x^3}{8x^2 + 3} $$ is **odd**.

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