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 ->  -> 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

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 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(9717)   (Show Source): You can put this solution on YOUR website!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```