Finding the linear functions from a function composition. This is a practice question in my book:

Find all linear functions $f$ such that
   $$(f \circ f)(x) = 4x+1.$$

Since linear functions are of the form
$$ax + b$$
I do this:
$$a(ax+b)+b\\
=a^2x+ab+b = 4x+1$$
If I solve the system
$$a^2 = 4\\
ab+b = 1$$
I get two solutions:
$$a = 2\\
b = \frac{1}{3}$$
However, I don't know what to do now. According to my book, the answer is that there are two functions:
$$f_1=2x+\frac{1}{3}\\
f_2=-2x-1$$
I think I see the relation between my two solutions and the first function in the answer. But I don't know what happened with the second function in the answer. How did my book get that answer?

You also have $a = -2$
Since $$ab + b = b(1 + a) = -b,$$
you get $b = -1$. This gives you the second solution
$y = -2x - 1$.
