Final answer:
The inverse of the function g(x) = ⌊x⌋ can be determined by identifying the values of x that satisfy specific conditions. In the case of g⁻¹({0}), the values of x that round down to 0 are between 0 and 1. For g⁻¹({-1, 0, 1}), the values of x that round down to -1, 0, or 1 are between -1 and 2. Finally, g⁻¹({x∣0
Step-by-step explanation:
(a) To find g⁻¹({0}), we need to find the values of x that satisfy the equation g(x) = 0. Since g(x) = ⌊x⌋, the values of x that round down to 0 are the numbers between 0 and 1. Therefore, g⁻¹({0}) = [0, 1).
(b) To find g⁻¹({-1, 0, 1}), we need to find the values of x that satisfy the equation g(x) = -1, 0, or 1. Since g(x) = ⌊x⌋, the values of x that round down to -1, 0, or 1 are the numbers between -1 and 2, inclusive. Therefore, g⁻¹({-1, 0, 1}) = [-1, 2].
(c) The notation {x∣0<x∞} represents all the values of x that are greater than 0. Since g(x) = ⌊x⌋, the values of x that round down to a number greater than 0 are all positive integers. Therefore, g⁻¹({x∣0<x∞}) = {1, 2, 3, ...}.