A vector space over R is a set V of objects (called vectors) together with two operations, addition and multiplication by scalars (real numbers), that satisfy the following 10 axioms. The axioms must hold for all vectors u, v,w in V and for all scalars α,β in R. (Closed under addition:) The sum of u and v, denoted u+v, is in V. (Closed under scalar multiplication:) The scalar multiple of u by α, denoted αu, is in V. (Addition is commutative:) u+v=v+u. (Addition is associative:) (u+v)+w=u+(v+w). (A zero vector exists:) There exists a vector 0 in V such that u+0=u. (Additive inverses exist:) For each u in V, there exists a v in V such that u+v=0. (We write v=−u.) (Scaling by 1 is the identity:) 1u=u. (Scalar multiplication is associative): α(βu)=(αβ)u. (Scalar multiplication distributes over vector addition:) α(u+v)=αu+αv. (Scalar addition is distributive:) (α+β)u=αu+βu. Let V be the set of functions f:R→R. For any two functions f, g in V, define the sum f+g to be the function given by (f+g)(x)=f(x)+g(x) for all real numbers x. For any real number c and any function f in V, define scalar multiplication cf by (cf)(x)=cf(x) for all real numbers x. Answer the following questions as partial verification that V is a vector space.