4. Sigmoid Function

Implement the sigmoid activation function.

$\sigma(x) = \frac{1}{1 + e^{-x}}$

The sigmoid maps any real number to the range $(0, 1)$, making it useful for binary classification outputs and gates in recurrent networks.

Properties to note:

• $\sigma(0) = 0.5$
• $\sigma(x) \to 1$ as $x \to +\infty$
• $\sigma(x) \to 0$ as $x \to -\infty$
• $\sigma'(x) = \sigma(x)(1 - \sigma(x))$

Example

Input:  x = 0
Output: 0.5

Input:  x = 2
Output: ≈ 0.8808

Constraints

• Your function should work on both scalars and NumPy arrays.