A Four-Function Functional Equation
Let the real -valued functions be defined on such that they satisfy the previous functional equation for all possible values of and in and is not a constant function.
If represents the largest possible value of and the smallest value of find Enter 666 if either or does not exist.
The answer is 0.
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1 solution
Relevant wiki: Functional Equations - Problem Solving
By making x = 0 in the functional equation, we get 1 + g ( y ) = k ( y ) for all values of y . So g ( y ) = k ( y ) − 1 . Substituting g ( y ) into the given functional equation, we get f ( x ) + k ( y ) − 1 = h ( x ) k ( y ) , that can be rewritten as f ( x ) − 1 = ( h ( x ) − 1 ) k ( y ) . ( ∗ ) If there is a number x 1 in [ 0 , 1 ] , such that h ( x 1 ) = 1 , then k ( y ) = h ( x 1 ) − 1 f ( x 1 ) − 1 , that would contradict the fact that k is not a constant function. That is why h ( x ) = 1 for all x in [ 0 , 1 ] . Then using ( ∗ ) , we obtain that f ( x ) = 1 for all values of x in [ 0 , 1 ] . Hence f ( 1 ) = 1 . Therefore, the answer to this question would be M − m = 1 − 1 = 0 .