Determine whether the set equipped with the given operations is a vector space. For those that are not vector spaces identify the vector space axioms that fail. The set of all real-valued functions f defined everywhere on the real line and such that f(14) = 0, with the operations(f+g)(x) = f(x) + g(x) (kf)(x) = kf(x)

White shapes and black shapes are used in a game. some of the shapes are circles. while the rest are squares. the ratio of the number of white shapes to the number of black shapes is 5: 11. the ratio of the number of white cicrles to the number of white squares is 3: 7. the ratio of the number of black circles to the number of black squres is 3: 8. work out what fraction of all the shapes are circles.