Inverse function of bi-exponential function?

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Homework Help Overview

The discussion revolves around finding the inverse function of a bi-exponential function represented as Y=A*exp(B*X)+C*exp(D*X). Participants explore the challenges associated with inverting such functions, particularly in contrast to single exponential functions.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to apply logarithmic methods used for single exponential functions to the bi-exponential case but expresses uncertainty about how to proceed. Some participants question the existence of an inverse function for certain forms of bi-exponential functions, while others suggest that domain restrictions might allow for an inverse.

Discussion Status

The discussion is active, with participants exploring different interpretations of the problem. Some guidance has been offered regarding the necessity of domain restrictions for the existence of an inverse, and the potential need for numerical or graphical methods to approximate solutions has been noted.

Contextual Notes

Participants mention specific constraints regarding the parameters of the bi-exponential function, including that A and C should be greater than 0, and B and D should be less than 0. The domain of X is specified as 0 to infinity.

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Homework Statement


What is the inverse function of a bi-exponential function like the following:
Y=A*exp(B*X)+C*exp(D*X)

Homework Equations


Y=A*exp(B*X)


The Attempt at a Solution


If it is a single exponential function, i can take log on both sides to get inverse function. But when it comes to a bi-exponential function, i really don't know how to do it. Thanks in advance.
 
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If y(x)=(1/2)(e^x+e^(-x)) (that's the cosh function), then y(1)=y(-1). It doesn't even have an inverse.
 
In fact, the bi-exponential equation i described here is a bi-exponential decay equation. The domain of X is 0 to inf. The parameter A and C should be greater than 0 and B, D should be less than 0. I have plotted a simulated data with R. It seems the inverse function of the bi-exponential function should be a multiple logarithmic function.
 
Ok, if you restrict the domain, then it can have an inverse. But I don't think there is a nice way to solve for x in terms of y except in special cases. You'll have to use graphical or numerical techiques to get an approximate solution. As you've noticed, logs don't get you there.
 
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