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)(0.7), f^(-1)(0.4), f^(-1)(0.5).
a. choose the correct answer below.
a. no horizontal line intersects the graph of f in more than one point.
b. a horizontal line intersects the graph of f in more than one point.
b. choose the correct description for f^(-1)(0.7) below.
a. f^(-1)(0.7), or approximately 30, represents the number of people who would have to be in the room so that the probability of two not sharing a birthday would be 0.7.
b. f^(-1)(0.7), or approximately 30, represents the number of people who would have to be in the room so that the probability of two sharing a birthday would be 0.7.

Explanation:

Step1: Recall inverse - function condition

A function has an inverse that is a function if and only if it is one - to - one. A one - to - one function passes the horizontal line test, meaning no horizontal line intersects the graph of the function in more than one point. Looking at the graph of \(y = f(x)\), we can see that no horizontal line intersects the graph of \(f\) in more than one point.

Step2: Interpret the inverse function

The function \(y = f(x)\) gives the probability that two people in the same room share a birthday as a function of the number of people in the room. The inverse function \(f^{-1}(p)\) gives the number of people in the room as a function of the probability \(p\) that two people share a birthday. So, \(f^{-1}(0.7)\) represents the number of people who would have to be in the room so that the probability of two sharing a birthday would be \(0.7\).

Answer:

a. A. No horizontal line intersects the graph of \(f\) in more than one point
b. The number of people who would have to be in the room so that the probability of two sharing a birthday would be \(0.7\)