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)
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 not sharing a birthday would be 0.7
choose the correct description for f^(-1)(0.4) below
a. f^(-1)(0.4), or approximately 20, represents the number of people who would have to be in the room so that the probability of two sharing a birthday would be 0.4
b. f^(-1)(0.4), or approximately 20, 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 0.4
choose the correct description for f^(-1)(0.5) below
a. f^(-1)(0.5), or approximately 23, represents the number of people who would have to be in the room so that the probability of two sharing a birthday would be 0.5
b. f^(-1)(0.5), or approximately 23, 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 0.5

Explanation:

Step1: Check one - to - one property

A function has an inverse that is a function if and only if it is one - to - one. Looking at the graph of \(y = f(x)\), as the number of people (the \(x\) - variable) increases, the probability that two people in the room share a birthday (the \(y\) - variable) is a strictly increasing function. That is, for any two distinct values \(x_1\) and \(x_2\) in the domain of \(f\), if \(x_1

Step2: Interpret \(f^{-1}(y)\)

The function \(y = f(x)\) gives the probability that two people in the room share a birthday as a function of the number of people \(x\) in the room. The inverse function \(x = f^{-1}(y)\) gives the number of people \(x\) in the room as a function of the probability \(y\) that two people in the room share a birthday.

• For \(f^{-1}(0.7)\): The value \(f^{-1}(0.7)\approx30\) means that when the probability that two people in the room share a birthday is \(0.7\), the number of people in the room is approximately \(30\).
• For \(f^{-1}(0.4)\): The value \(f^{-1}(0.4)\approx20\) means that when the probability that two people in the room share a birthday is \(0.4\), the number of people in the room is approximately \(20\). So the correct option for \(f^{-1}(0.4)\) is A.
• For \(f^{-1}(0.5)\): The value \(f^{-1}(0.5)\approx23\) means that when the probability that two people in the room share a birthday is \(0.5\), the number of people in the room is approximately \(23\). So the correct option for \(f^{-1}(0.5)\) is A.

Answer:

a. The function \(f\) is strictly increasing, so it is one - to - one and has an inverse that is a function.
b.

• \(f^{-1}(0.7)\approx30\) represents the number of people who would have to be in the room so that the probability of two sharing a birthday would be \(0.7\).
• A. \(f^{-1}(0.4)\), or approximately 20, represents the number of people who would have to be in the room so that the probability of two sharing a birthday would be \(0.4\).
• A. \(f^{-1}(0.5)\), or approximately 23, represents the number of people who would have to be in the room so that the probability of two sharing a birthday would be \(0.5\).