+1768 Let $f(x) = px + q$, where $p$ and $q$ are real numbers. Find $p+q$ if $f(f(f(x))) = 64x - 105 - 30x + 70 + 32x - 12$.
+129850 f (f ( f(x))) = 66x - 47
f (f (x) ) = p(px + q) + q = p^2x + pq + q
f ( f ( f (x)) ) = p (p^2x + pq + q) + q = p^3x + p^2q + pq + q
So
p^3x + p^2q + pq + q = 66x - 47
p^3 = 66
p= 66^(1/3)
And
p^2q + pq + q = -47
66^(2/3)q + 66^(1/3)*q + q = -47
q (66^(2/3) + 66^(1/3) + 1) = -47
q = -47 / ( 66^(2/3) + 66^(1/3) + 1)
p + q = [ 66^(1/3) ( 66^(2/3) + 66^(1/3) + 1) - 47] / [ 66^(2/3) +66^(1/3) + 1 ] =
[ 66^(2/3) + 66^(1/3) + 19 ] / [66^(2/3) + 66^(1/3) + 1 ] ≈ 1.842.....
