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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?

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    If $a^2=4$, $a$ can be either $2$ or $-2$.2012-07-08

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