Question

Let V be the set of all real-valued functions defined on R. Define f⊕g by (f⊕g)(x)=f(x)+g(x) and define c⊙f by (c⊙f)(x)=f(x+c)

Answer 1

The question is about defining function operations in terms of vector addition and horizontal function shifts, similar to one-dimensional vector addition and algebraic function translations.

Step-by-step explanation:

The student is inquiring about the operations on the set V of real-valued functions defined on R. The operation (f⊕g)(x)=f(x)+g(x) defines addition of functions which is similar to vector addition in one dimension, reflecting the commutative property as with vectors A+B = B+A. The operation (c⊕f)(x)=f(x+c) describes a horizontal shift of the function, akin to translating a function by a distance d in algebra, and is similar to wave functions' translations in physics.