Final answer:
The set of all real-valued functions V with addition and scalar multiplication where cf(x)=f(cx) does not form a vector space because it violates vector space axioms, particularly the distributive property of scalar multiplication over vector addition.
Step-by-step explanation:
The set of all real-valued functions V combined with the usual operation of addition and the defined scalar multiplication where cf(x) = f(cx) does not form a vector space. To be considered a vector space, a set must satisfy certain axioms under the operations of vector addition and scalar multiplication. The basic properties these operations must adhere to include associativity and commutativity of vector addition, the distributive properties of scalar multiplication over vector addition and vector addition over scalar addition, and lastly, scalar multiplication by a scalar quantity which affects only the magnitude and possibly direction of a vector, described by cA.
In this scenario, the given definition of scalar multiplication cf(x) = f(cx) violates the axiom that requires the operation c(A + B) = cA + cB. Since we are modifying the argument of the function rather than scaling its output, it breaks this distributive property and hence, the set V under these operations cannot be considered a vector space.