The function given is:

$C(x) = \frac{100}{1 + e^{-0.5(x - 10)}}$

This is a sigmoid function, often used to model growth processes or probability distributions.

Analysis of the Function:

1. Domain: The function is defined for all $x$, but you're interested in the interval $x = 0$ to $x = 20$.

2. Behavior:

   • As $x$ approaches 0, $C(x)$ will start near 0.
   • As $x$ increases, the function gradually increases and approaches 100 asymptotically.

3. Key points:

   • Midpoint: At $x = 10$, the exponent $-0.5(x - 10)$ becomes 0, so $C(10) = \frac{100}{1 + e^0} = \frac{100}{2} = 50$.
   • Growth rate: The rate at which $C(x)$ approaches 100 is controlled by the coefficient 0.5 in the exponent.

4. Values at Specific Points:

   • $C(0) = \frac{100}{1 + e^{5}}$
   • $C(20) = \frac{100}{1 + e^{-5}}$

Would you like me to compute specific values or plot this function for better visualization?

Related Questions:

1. What is the value of $C(x)$ at $x = 0$, $x = 5$, $x = 10$, $x = 15$, and $x = 20$?
2. How does changing the coefficient in the exponent (e.g., from 0.5 to 1) affect the shape of the sigmoid curve?
3. What happens to $C(x)$ if we extend the interval beyond $x = 20$?
4. How does the midpoint $x = 10$ relate to the function's inflection point?
5. What are some real-world applications of this sigmoid function?

Tip: Sigmoid functions like this one are often used in logistic regression models and neural networks to map predictions to probabilities.