1. **State the problem:** We need to analyze the function $$f(x) = \sin^3\left(\cos\sqrt{x^3 + 2x^2}\right)$$ and understand its behavior. 2. **Understand the function:** The function is a composition of several functions: - Inside the square root: $$x^3 + 2x^2$$ - Then the cosine of the square root: $$\cos\sqrt{x^3 + 2x^2}$$ - Then the sine cubed of that: $$\sin^3(\cdot)$$ 3. **Domain considerations:** The expression inside the square root must be non-negative: $$x^3 + 2x^2 \geq 0$$ Factor: $$x^2(x + 2) \geq 0$$ Since $$x^2 \geq 0$$ always, the inequality depends on $$x + 2 \geq 0$$, so $$x \geq -2$$ 4. **Range of inner functions:** - $$\sqrt{x^3 + 2x^2}$$ is real and non-negative for $$x \geq -2$$. - $$\cos(\text{anything real})$$ ranges between $$-1$$ and $$1$$. - $$\sin^3(y) = (\sin y)^3$$ ranges between $$-1$$ and $$1$$. 5. **Summary:** The function $$f(x)$$ is defined for $$x \geq -2$$ and its values lie between $$-1$$ and $$1$$. 6. **No further simplification is possible without specific values or derivatives.**