Question 1180625
**Probability Function:**

The probability function P(x) that a fan in attendance on day x will receive a baseball cap is:

P(x) = (x + 5) / (2x + 3)

where:

*   x = the game number (x = 1 for the first game, x = 2 for the second game, and so on)
*   x + 5 = the number of baseball caps given away on day x
*   2x + 3 = the number of fans attending on day x

**Domain:**

In practical terms, the domain of the probability function is the set of positive integers representing the game number. Since the number of caps and attendees cannot be negative, and the game number must be a whole number, the domain is:

Domain: {x | x is an integer, x ≥ 1}

**Range:**

The range of the probability function represents the possible probabilities of a fan receiving a baseball cap. Since the number of caps given away is always less than the number of fans attending, the probability will always be between 0 and 1. As the game number increases, the probability approaches 1/2 since the number of caps and the number of attendees both grow linearly with x, but the number of attendees grows twice as fast.

Range: {P(x) | 0 < P(x) < 1}