School Bus Assignment: Function, Inverse and Composition
A school in Guwahati runs four buses B1, B2, B3 and B4 for five students: Ananya, Bikash, Chiranjit, Devika and Eshan. The transport register records the assignment f as Ananya→B2, Bikash→B1, Chiranjit→B2, Devika→B4 and Eshan→B1. A separate rule g gives the seating capacity of each bus: B1→40, B2→32, B3→48 and B4→28. The coordinator says f is 'a function from students to buses' and asks whether its inverse can recover a student from a bus number. A parent also wants to know the capacity that applies to each student through g∘f. A new student Farhan is to be added. B1 and B2 have no spare seats on their routes; B3 and B4 have 5 and 3 spare seats respectively. The coordinator wonders whether Farhan's bus can be chosen so that the extended assignment remains a function and whether the inverse relation from buses to students would then be a function.
1. Is f a function? Is it injective? Is f^{-1} a function from the set of buses to the set of students? Justify each answer.
Read the worked answer
f is a function because every student in the domain is assigned exactly one bus. It is not injective: Ananya and Chiranjit both map to B2, and Bikash and Eshan both map to B1. For f^{-1} to be a function from buses to students, each bus would need at most one preimage. But B2 has two preimages and B1 has two, so f^{-1} contains (B2, Ananya) and (B2, Chiranjit), for example. Also B3 has no preimage, so f^{-1} would not even be defined on all four buses. Thus f^{-1} is only a relation, not a function.
2. Compute g∘f for each student and state its domain and codomain. Does g∘f answer the parent's question about individual seats?
Read the worked answer
The composition is defined because f maps students into the set of buses, which is the domain of g. Computing: Ananya→B2→32, Bikash→B1→40, Chiranjit→B2→32, Devika→B4→28, Eshan→B1→40. So g∘f = {(Ananya,32), (Bikash,40), (Chiranjit,32), (Devika,28), (Eshan,40)}. Its domain is the five students; a suitable codomain is the set of capacities {28,32,40,48}, or the natural numbers. It gives the capacity of the bus assigned to each student, not a guaranteed individual seat. Since a bus can carry up to its capacity, the value is a maximum, and it is not injective because capacities repeat.
3. Farhan can be assigned only to B3 or B4. If the extended relation f' on the six students must remain a function, what are the options? If f'^{-1} must also be a function, what further restriction arises?
Read the worked answer
To keep f' a function, Farhan must be given exactly one bus, so the valid options are Farhan→B3 or Farhan→B4. Both preserve the single-output rule. For f'^{-1} to be a function, no bus may have two students assigned to it. But B1 already has Bikash and Eshan, and B2 already has Ananya and Chiranjit, so f'^{-1} cannot be a function regardless of Farhan's choice. The only way to satisfy the inverse-function requirement would be to reassign existing students so that every used bus has at most one student, which the current register does not do.