Let f : R → R is a strictly increasing function and f − 1 is its inverse function. It satisfies: f ( x 1 ) + x 1 = a; f − 1 ( x 2 ) + x 2 = aWhat is the value of x 1 + x 2 ?

Question

Let   f  :      R    →      R   is a strictly increasing function and       f          −      1       is its inverse function. It satisfies:  f  (      x    1    )  +      x    1    =  a;       f          −      1        (      x    2    )  +      x    2    =  aWhat is the value of       x    1    +      x    2  ?

Glenn Ware · Accepted Answer

It seems the following. Since the function   f is strictly increasing, then the functions       f          −      1       and   g  (  x  )  =  f  (  x  )  +  x are strictly increasing too. We have   a  =  g  (      x    1    )  =  g  (      f          −      1        (      x    2    )  ). Then       x    1    =      f          −      1        (      x    2    ) and       x    1    +      x    2    =      f          −      1        (      x    2    )  +      x    2    =  a.

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