If p(3x) = x^2 and....

Question

If p(3x) = x² and p(2x) = x, is it possible that p is a linear function from the polynomials to the polynomials?  Explain your answer.  (Hint:  Fractions are scalors too)

Answer 1

If I am interpreting the question correctly it is asking do there exist constants (scalars) a and b such that

p(f(x)) = af(x) + b,

 

Since p(3x) = a(3x) + b = x²  then 3a = x  therefore a=x/3 which is not a scalar.

 

Is this correct or am I misinterpreting the question?