Question

the given graph represents the probability that two people in the same room share a birthday as a function of the number of people in the room. call the function f. complete parts a and b. a. explain why f has an inverse that is a function. b. describe in practical terms the meanings of f⁻¹(0.25), f⁻¹(0.2), f⁻¹(0.4).

Explanation:

a. A function has an inverse that is a function if it is one-to-one (injective), meaning no two distinct inputs map to the same output. The graph of \( y = f(x) \) is strictly increasing (as the number of people increases, the probability only rises), so it passes the horizontal line test (no horizontal line intersects the graph more than once).
b. The inverse function \( f^{-1}(y) \) gives the number of people needed for the probability of a shared birthday to be \( y \). Thus, \( f^{-1}(0.25) \) is the number of people for a 25% probability, \( f^{-1}(0.2) \) for 20%, and \( f^{-1}(0.4) \) for 40%.

Answer:

a. \( f \) is strictly increasing, so it is one-to-one and has an inverse function.
b. \( f^{-1}(0.25) \): Number of people needed for a 25% probability of a shared birthday; \( f^{-1}(0.2) \): Number of people needed for a 20% probability; \( f^{-1}(0.4) \): Number of people needed for a 40% probability.