Let , where and are real numbers. Find if .

Let , where  and  are real numbers. Find  if 

f(f(f(x)))   = (f o f o f)(x)

We first want to put f into f...this gives

p(px + q) + q = p²x + pq + q

Now....we want to put this result into f, again...so we have

p[ p²x + pq + q] + q =  p³x + p²q + pq + q     and setting this to 8x + 21, we have

p³x + p²q + pq + q  = 8x + 21   equating coefficients .....   p³ = 8   so p = 2  ....so we have

8x + 4q + 2q + q  = 8x + 21     subtract 8x from both sides

4q + 2q + q = 21     simplify

7q = 21      divide both sides by 7

q = 3

So .... p + q =  2 + 3   = 5