1. **State the problem:** Find the function and analyze the expression for $$y = \left( \frac{x^3 - 1}{2x + 1} \right)^4$$ 2. **Formula and rules:** This is a composite function where a rational function is raised to the fourth power. 3. **Intermediate work:** - The numerator is $x^3 - 1$, which can be factored as $$(x - 1)(x^2 + x + 1)$$ - The denominator is $2x + 1$. - The entire fraction is raised to the power 4. 4. **Explanation:** - The function is defined for all $x$ such that $2x + 1 \neq 0$, i.e., $x \neq -\frac{1}{2}$. - The fourth power ensures the output is always non-negative since any real number to an even power is non-negative. 5. **Final expression:** $$y = \left( \frac{(x - 1)(x^2 + x + 1)}{2x + 1} \right)^4$$ This function can be graphed to observe behavior near $x = -\frac{1}{2}$ (vertical asymptote) and zeros at $x=1$.