Let f(x)=px+q, where p and q are real numbers. Find p+q if f(f(f(x)))=8x+21. Thank you.

Question

I got this p^3 +p^2q + pq + q = 29
Would you try to solve that? if yes, then how?

Answer 1

f(f(x)] = p(px+q) + q

f[f(f(x)) = p[p(px+q) + q] + q
= p^2(px+q) + pq + q

p^3 x + p^2 q + pq + q = 8 x + 21

= (p^3-8)x + p^2 q + pq + q = 21
this can not be a function of x or it would not be constant 21
so
p^3-8 = 0
p = 2

4 q + 2q + q = 21
q = 3