Final answer:
The question is whether the provided function is a homomorphism from the group Z_4 + Z_8 to Z_8, which requires it to preserve the group operation. Without additional information, it cannot be conclusively determined if the function is a homomorphism.
Step-by-step explanation:
The question asks whether the given function f((a,b)) \rightarrow 3b \mod 8 is a homomorphism from the group Z_4 + Z_8 to the group Z_8. A function between two groups is a homomorphism if it preserves the group operation, meaning that f(x+y) = f(x) \cdot f(y) for all elements x and y in the first group.
In this case, the operation in the domain group Z_4 + Z_8 is addition modulo 4 for the first component and addition modulo 8 for the second component. In the codomain Z_8, the operation is addition modulo 8.
However, the provided function does not preserve the group operations based on the information given; hence it cannot be concluded that it is a homomorphism without additional context or a clearer definition of the function f and how it operates on pairs (a, b).