1. **State the problem:** We need to simplify and understand the function given as $Y = \frac{1}{1 + \sqrt{x}}$. 2. **Formula and rules:** The function involves a square root in the denominator. To simplify expressions like this, we often rationalize the denominator by multiplying numerator and denominator by the conjugate. 3. **Intermediate work:** Multiply numerator and denominator by $1 - \sqrt{x}$: $$ Y = \frac{1}{1 + \sqrt{x}} \times \frac{1 - \sqrt{x}}{1 - \sqrt{x}} = \frac{1 - \sqrt{x}}{(1 + \sqrt{x})(1 - \sqrt{x})} $$ 4. Simplify the denominator using the difference of squares formula: $$ (1 + \sqrt{x})(1 - \sqrt{x}) = 1 - (\sqrt{x})^2 = 1 - x $$ 5. So the simplified form is: $$ Y = \frac{1 - \sqrt{x}}{1 - x} $$ 6. **Explanation:** Rationalizing the denominator helps to remove the square root from the denominator, making the expression easier to work with in further calculations or evaluations. **Final answer:** $$ Y = \frac{1 - \sqrt{x}}{1 - x} $$