If f(f(x))=4x+5 what does f(x)=?

Question

Your answer must be a linear function :-)

Answer 1

Let f(x) = ax+b, then f(f(x)) = f(ax+b) = a(ax+b)+b = a²x + (ab+b) = 4x +5

a² = 4, (a+1)b = 5

So, a = 2, b = 5 / (2+1) = 5/3

Or a = -2, b = 5 / (-2+1) = -5

There are 2 solutions:

f₁(x) = 2x + 5/3 and f₂(x) = -2x - 5

Answer 2

f(x) = 2x + (5/3)