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.

Algebra ->  Functions -> 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.      Log On


   

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) About Me  (Show Source):
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. Equate terms with "x" in it, as well as constants:

and

Substitute b with a^2 in the second equation:

. Solving for a, (here, denotes the golden ratio)

Since b = a^2,

Therefore,