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 626326: 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 Edwin McCravy(20056) About Me  (Show Source):
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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.
        f(x) = a·x + b

      f(f(x)) = a·f(x) + b = a(a·x + b) + b = a²·x + ab + b

      f(f(x)) = b·x + a given.  Therefore for all x

a²·x + ab + b = b·x + a

Let x = 1 (allowed since true for all x

a²·1 + ab + b = b·1 + a

  a² + ab + b = b + a

      a² + ab = a

Divide through by a  (allowed since a is positive)

       a + b = 1

Edwin