Prove that each of the following is a homomorphism and descr

Prove that each of the following is a homomorphism, and describe its kernel.

Solution

phi(f1+f2) = (f1+f2)(0) (by definition of phi)
Or, phi(f1+f2) = f1(0) + f2(0) (by property of functions)-----(1)

phi(f1.f2) = (f1.f2)(0) (by definition of phi)
Or, phi(f1.f2) = f1(0).f2(0) (by property of functions)------(2)

From (1) and (2), phi is a homomorphism.


Kernel:
K = {x E F(R) : phi(x) = 0} (by definition of kernel)


Range:
Range = {phi(f) : f E F(R)}
Or, range = {f(0)} (by definition of phi)