1. **State the problem:** We want to analyze the function $$f(x) = \frac{x^{\frac{3}{2}} + 1}{x^{\frac{1}{2}} - x}$$ and simplify or understand its behavior. 2. **Rewrite the function using radicals:** Recall that $$x^{\frac{1}{2}} = \sqrt{x}$$ and $$x^{\frac{3}{2}} = x \cdot \sqrt{x}$$. So, $$f(x) = \frac{x \sqrt{x} + 1}{\sqrt{x} - x}$$. 3. **Factor the denominator:** Notice that $$x = (\sqrt{x})^2$$, so $$\sqrt{x} - x = \sqrt{x} - (\sqrt{x})^2 = \sqrt{x}(1 - \sqrt{x})$$. 4. **Rewrite the numerator:** The numerator is $$x \sqrt{x} + 1 = (\sqrt{x})^3 + 1$$. 5. **Recognize the numerator as a sum of cubes:** Since $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$, with $$a = \sqrt{x}$$ and $$b = 1$$, we have $$ (\sqrt{x})^3 + 1^3 = (\sqrt{x} + 1)(x - \sqrt{x} + 1) $$. 6. **Rewrite the function using these factorizations:** $$f(x) = \frac{(\sqrt{x} + 1)(x - \sqrt{x} + 1)}{\sqrt{x}(1 - \sqrt{x})}$$. 7. **Simplify the denominator:** Note that $$1 - \sqrt{x} = -(\sqrt{x} - 1)$$, so $$f(x) = \frac{(\sqrt{x} + 1)(x - \sqrt{x} + 1)}{\sqrt{x} \cdot -(\sqrt{x} - 1)} = - \frac{(\sqrt{x} + 1)(x - \sqrt{x} + 1)}{\sqrt{x}(\sqrt{x} - 1)}$$. 8. **Final simplified form:** $$f(x) = - \frac{(\sqrt{x} + 1)(x - \sqrt{x} + 1)}{\sqrt{x}(\sqrt{x} - 1)}$$. This form helps analyze the function's behavior, domain restrictions (e.g., $$x \geq 0$$ and $$\sqrt{x} \neq 1$$), and asymptotes. **Answer:** $$f(x) = - \frac{(\sqrt{x} + 1)(x - \sqrt{x} + 1)}{\sqrt{x}(\sqrt{x} - 1)}$$