The given function is:

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

This function is known as the logistic function, which is commonly used in probability and machine learning (sigmoid function). It has the following characteristics:

1. Asymptotic Behavior:

   • As $x \to -\infty$, $f(x) \to 1$.
   • As $x \to \infty$, $f(x) \to 0$.

2. Monotonicity:

   • It is a decreasing function, meaning it continuously decreases as $x$ increases.

3. Point of Inflection:

   • At $x = 0$, $f(x) = \frac{1}{1 + e^0} = \frac{1}{2}$.
   • The function is symmetric about $(0, 0.5)$.

From the given options in the image, the correct graph should:

• Start near $y = 1$ for negative $x$.
• Decrease smoothly and pass through $(0, 0.5)$.
• Approach $y = 0$ for large positive $x$.

The correct answer is option (H).

Would you like a step-by-step explanation of why this choice is correct?

Related Questions:

1. What happens if we change the function to ( f(x) = \frac{1}{1 +