Inverse of the function − log ⁡ ( 1 − [ 1 − e −...

Violet Woodward

Violet Woodward

Answered

2022-07-18

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

Answer & Explanation

decoratesuw

decoratesuw

Expert

2022-07-19Added 11 answers

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

Raegan Bray

Expert

2022-07-20Added 1 answers

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)

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