QUESTION
Please spend extra effort to write up this problem’s solution as an exposition that can be read and understood by a beginning algebra student. That student knows function notation and standard properties of polynomials (as taught in a high school algebra course).
(a) Find all polynomials f that satisfy the equation: f(x + 2) = f(x) + 2 for every real number x.
(b) Find all polynomials g that satisfy the equation: g(2x) = 2g(x) for every real number x.
(c) The problems above are of the following type: Given functions H and J, find all polynomials Q that satisfy the equation: J(Q(x)) = Q(H(x)) for every x in S where S is a subset of real numbers. In parts (a) and (b), we have J = H and S is all real numbers, but other scenarios are also interesting. For example, the choice J(x) = 1/(x − 1) and H(x) = 1/(x + 1), generates the Find all polynomials Q that satisfy the equation: 1/(Q(x) − 1) = Q(1/(x + 1)) for every real number x such that those denominators are nonzero. Is this one straightforward to solve?
(d) Make your own choice for J and H, formulate the problem, and find a solution. Choose J and H to be non-trivial, but still simple enough to allow you to make good progress toward a solution