If \(f\left( x \right) =ax+b\), where \(a\) and \(b\) are real numbers, and \(f\left( f\left( f\left( x \right) \right) \right) =8x+21\), what is \(a+b\)?
f(x) = ax + b
f(f (x)) = a ( ax + b) + b = a^2x + ab + b
f (f ( f(x) ) ) = a^2 ( ax + b) + ab + b = a^3x + a^2b + ab + b = 8x + 21
Equating terms a^3 = 8 ⇒ a =2
And
a^2b + ab + b = 21
(2)^2 b + 2b + b =21
4b + 2b + b = 21
7b = 21
b =3
a + b = 2 + 3 = 5
