Precalc question

Question

(a) Let f(x) be a linear function, f(x) = ax + b for constants a and b. Show that f(f(x)) is a linear function.

(b) Find a function g(x) such that g(g(x)) = 6x − 8.

Answer 1

(a) f(x) = ax + b => f(f(x)) = a(ax + b) + b = a^2(x) + ab + b = a^2(x) +(b+1)a = Ax + B for A = a^2 and B = (a+1)b

so f(f(x)) = Ax + B, which is also linear.

b) = g(g(x)) = 6x - 8; from (a) A = 6, B = - 8, a = sqrt(6), [sqrt(6)+1](b) = B = -8, or b = -8/(sqrt(6) +1)., so g(x) = sqrt(6)*x + (-8/(sqrt(6) +1))