1. **State the problem:** We need to analyze the function $$y = -4x^5 + \sqrt{x^3} + 3x^2 - 7$$ and understand its components. 2. **Rewrite the function:** Recall that $$\sqrt{x^3} = (x^3)^{\frac{1}{2}} = x^{\frac{3}{2}}$$. So the function becomes: $$y = -4x^5 + x^{\frac{3}{2}} + 3x^2 - 7$$ 3. **Explain the terms:** - The term $$-4x^5$$ is a polynomial term with degree 5. - The term $$x^{\frac{3}{2}}$$ is a power function with fractional exponent. - The term $$3x^2$$ is a polynomial term with degree 2. - The constant term is $$-7$$. 4. **Domain considerations:** Since $$x^{\frac{3}{2}} = (\sqrt{x})^3$$, the square root requires $$x \geq 0$$ for real values. 5. **Summary:** The function is defined for $$x \geq 0$$ and combines polynomial and fractional power terms. No further simplification is possible without specific values or operations requested.