Question 627113
*[tex \LARGE f(x) = ax + b]


*[tex \LARGE f(f(x)) = f(ax+b) = a(ax + b) + b = bx + a]


*[tex \LARGE a^2x + ab + b = bx + a]. Equate terms with "x" in it, as well as constants:


*[tex \LARGE a^2 = b] and *[tex \LARGE ab + b = a]


Substitute b with a^2 in the second equation:


*[tex \LARGE a(a^2) + a^2 = a \Rightarrow a^2 + a = 1 \Rightarrow a^2 + a - 1 = 0]. Solving for a, *[tex \LARGE a = \frac{1 + \sqrt{5}}{2} = \phi] (here, *[tex \LARGE \phi] denotes the golden ratio)


Since b = a^2, *[tex \LARGE b = \phi^2 = \phi + 1 = \frac{3 + \sqrt{5}}{2}]


Therefore, *[tex \LARGE a + b = \frac{1 + \sqrt{5}}{2} + \frac{3 + \sqrt{5}}{2} = 2 + \sqrt{5}]