1. **State the problem:** We are given the function $$f(x) = x^{\frac{2}{3}} + \sqrt{3 - x^{2}} \sin(16\pi x)$$ and we want to understand its behavior, such as domain and key features. 2. **Identify the domain:** The function has two parts: $$x^{\frac{2}{3}}$$ and $$\sqrt{3 - x^{2}} \sin(16\pi x)$$. - For $$x^{\frac{2}{3}}$$, since the exponent is $$\frac{2}{3}$$, which is the cube root squared, it is defined for all real $$x$$. - For $$\sqrt{3 - x^{2}}$$, the expression inside the square root must be non-negative: $$3 - x^{2} \geq 0 \implies x^{2} \leq 3 \implies -\sqrt{3} \leq x \leq \sqrt{3}$$. Therefore, the domain of $$f(x)$$ is $$[-\sqrt{3}, \sqrt{3}]$$. 3. **Explain the components:** - $$x^{\frac{2}{3}}$$ is a smooth function that grows as $$|x|^{2/3}$$. - $$\sin(16\pi x)$$ is a highly oscillatory function with period $$\frac{1}{8}$$. - The factor $$\sqrt{3 - x^{2}}$$ acts as an amplitude envelope, zero at $$x=\pm \sqrt{3}$$ and maximum at $$x=0$$. 4. **Summary:** The function combines a smooth increasing term and an oscillatory term with amplitude limited by $$\sqrt{3 - x^{2}}$$. No further simplification or closed form is possible. **Final answer:** The domain of $$f(x)$$ is $$[-\sqrt{3}, \sqrt{3}]$$ and the function is defined as given with oscillations modulated by $$\sqrt{3 - x^{2}}$$.