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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?

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up vote 4 down vote accepted

The system of equations $$ a^2=4\\ ab+b=1 $$

has two solutions; the first equation says $a=\pm 2$. Thus the second equation takes two forms:

$$ 2b+b=1\Rightarrow b=\frac{1}{3} $$ and

$$ -2b+b=1\Rightarrow b=-1 $$

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