SOLUTION: Let f(x)= ax+b, where a and b are positive numbers, and assume that the following equation holds for all values of x: f(f(x))= bx+a. Find a+b.
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Question 627113: Let f(x)= ax+b, where a and b are positive numbers, and assume that the following equation holds for all values of x: f(f(x))= bx+a. Find a+b.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
 = ax + b)
) = f(ax+b) = a(ax + b) + b = bx + a)
. Equate terms with "x" in it, as well as constants:
and 
Substitute b with a^2 in the second equation:
 + a^2 = a \Rightarrow a^2 + a = 1 \Rightarrow a^2 + a - 1 = 0)
. Solving for a, 
(here, 
denotes the golden ratio)
Since b = a^2, 
Therefore, 
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