Inverse of the function − log ⁡ ( 1 − [ 1 − e − x α ] β )I have a function as follows, I would like to get the inverse of this function. What is the inverse of f ( x )? y = f ( x ) = − log ⁡ ( 1 − [ 1 − e − x α ] β )Is my answer correct? f − 1 ( x ) = ( − log ⁡ ( 1 − ( 1 − exp − x ) 1 / β ) ) 1 / α

Question

Inverse of the function   −  log  ⁡  (  1  −  [  1  −      e          −              x        α                  ]    β    )I have a function as follows, I would like to get the inverse of this function. What is the inverse of   f  (  x  )?  y  =  f  (  x  )  =  −  log  ⁡  (  1  −  [  1  −      e          −              x        α                  ]    β    )Is my answer correct?      f          −      1        (  x  )  =  (  −  log  ⁡  (  1  −  (  1  −      exp          −      x            )          1              /            β        )      )          1              /            α

decoratesuw · Accepted Answer

Since you have already done the first step ..  y  =  f  (  x  )  =  −  log  ⁡  (  1  −  [  1  −  exp  ⁡  (  −      x    α    )      ]    β    )    ⟹    −      y    =  log  ⁡  (  1  −  [  1  −      e          −              x                  α                          ]          β        )    ⟹        e          −      y        =  1  −  [  1  −      e          −              x                  α                          ]          β        )    ⟹    −      e          −      y        +  1  =  [  1  −      e          −              x                  α                          ]          β          ⟹    [  1  −      e          −      y            ]                  1        β              −  1  =  −      e          −              x                  α                    which turns out to be      e          −              x                  α                      =  1  −  [  1  −      e          −      y            ]                  1        β            . Taking   log on both sides we get  −      x          α        =  log  ⁡  (  1  −  [  1  −      e          −      y            ]                  1        β              )    ⟹    l  o  g      1          (      1      −      [      1      −              e                  −          y                            ]                              1            β                              )        =      x    α  Hence      f          −      1        (  x  )  =            (      l      o      g              1                  (          1          −          [          1          −                      e                          −              x                                            ]                                          1                β                                              )                    )                      1        α

Raegan Bray · Accepted Answer

Multiply both sides by   −  1, exponentiate both sides, subtract 1 from both sides, multiply both sides by   −  1, raise both sides to the       1    β   power, subtract 1 from both sides, multiply both sides by   −  1, take natural logarithm of both sides, multiply both sides by   −  1, raise both sides to       1    α   power.This solves for   x in terms of   y, which gives you the inverse function in terms of   y(Note that you will need the       1    β   and       1    α   powers to be defined)