How do I calculate the inverse function of this function?I have this function: f ( x ) = 1 + ln ⁡ ( x ) 1 − ln ⁡ ( x ) And i should calculate f − 1 ( x )I am not really sure how to proceed but I think that the first step would be to have x alone, how do I achieve that?

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How do I calculate the inverse function of this function?I have this function:  f  (  x  )  =            1      +      ln      ⁡      (      x      )              1      −      ln      ⁡      (      x      )      And i should calculate       f          −      1        (  x  )I am not really sure how to proceed but I think that the first step would be to have x alone, how do I achieve that?

Jonathan Bailey · Accepted Answer

Let       f    (    x    )    =                  1        +        ln        ⁡        x                    1        −        ln        ⁡        x              =          y      1                ⟹              f              −        1              (    y    )    =    x  Applying Componendo &amp;amp; Dividendo,  ln  ⁡  x  =                    y        −        1                    y        +        1                      ⟹        x    =          e                        (                                                    y                −                1                                            y                +                1                                              )                     which is       f          −      1        (  y  )

daufleguos · Accepted Answer

Note, that when inversing a function you should also determine the set over which the inverse function is defined.Set   y  =  f  (  x  ) and solve for   x:  y  =            1      +      ln      ⁡      x              1      −      ln      ⁡      x        ⇒  y  (  1  −  ln  ⁡  x  )  =  1  +  ln  ⁡  x  ⇒  y  −  1  =  y  (  ln  ⁡  x  )  +  ln  ⁡  xwhich gives  (  y  −  1  )  =  (  y  +  1  )  ln  ⁡  x  ⇒  ln  ⁡  x  =            y      −      1              y      +      1      which gives by exponentiating both sides to the power   e      e          ln      ⁡      x        =      e                            y          −          1                          y          +          1                    which reduces to  x  =      e                            y          −          1                          y          +          1                    Therefore      f          −      1        (  y  )  =      e                            y          −          1                          y          +          1                    Note, also that this holds for all   y  ∈      R    ∖  {  −  1  }

How do I calculate the inverse function of this function? f(x)=(1+\ln(x))/(1-\ln(x))

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