Find the linear equations $f$ such that $(f\circ f)(x) = 4x+1$
If $f$ is linear, then it has the form $f(x) = ax+b$.
Since $f(f(x)) = a(ax+b)+b=a^2x+ab+b=4x+1$, we get the system
$a^2 = 4$ $ab+b = 1$
The solutions I got are $a=2$ and $b = \frac{1}{3}$.
So one of these linear functions is $f(x) = 2x+\frac{1}{3}$
The book says that the other one is $f(x) = -2x-1$, but, how did it get it?