There are two functions f : Q → Q ∖ { − 1 } and g : Q → Q ∖ { 1 }. g ( x ) = f ( x ) f ( x ) + 1 .Prove that if there is a inverse function for f, then g is also inverse.

Question

There are two functions   f  :      Q    →      Q    ∖  {  −  1  } and   g  :      Q    →      Q    ∖  {  1  }.  g  (  x  )  =            f      (      x      )              f      (      x      )      +      1        .Prove that if there is a inverse function for   f, then g is also inverse.

Jaylee Dodson · Accepted Answer

Let   h  (  x  )  =            x              x        +        1            , so  g  (  x  )  =  h  (  f  (  x  )  )  f  ,  h  have inverse function, so is   g.en       g          −      1        (  x  )  =      f          −      1        (      h          −      1        (  x  )  )

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