Inverse function of bi-exponential function?

In summary, the conversation discusses the inverse function of a bi-exponential function and how it differs from a single exponential function. It is noted that the bi-exponential function is a bi-exponential decay equation with a domain of 0 to infinity, and that the inverse function may be a multiple logarithmic function. However, it is also mentioned that there is no simple solution to solving for x in terms of y and that graphical or numerical techniques may be needed for an approximate solution.
  • #1
f_coco
2
0

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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  • #2
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.
 
  • #3
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.
 
  • #4
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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What is an inverse function?

An inverse function is a mathematical operation that "undoes" another function. It takes the output of a function as its input and produces the original input value as its output.

What is a bi-exponential function?

A bi-exponential function is a mathematical function that can be described by an equation of the form f(x) = a * exp(bx) + c * exp(dx), where a, b, c, and d are constants. It is often used to model growth or decay processes in science and engineering.

Why is it important to find the inverse function of a bi-exponential function?

Finding the inverse function of a bi-exponential function allows us to "undo" the original function and solve for the original input value. This is useful in many applications, such as solving for initial conditions or determining the time at which a certain value was reached.

How do you find the inverse function of a bi-exponential function?

To find the inverse function of a bi-exponential function, we can use the process of algebraically solving for the original input variable. This involves rearranging the equation and isolating the original input variable on one side of the equation.

Are there any restrictions or limitations when finding the inverse function of a bi-exponential function?

Yes, there are certain restrictions and limitations when finding the inverse function of a bi-exponential function. For example, the function must be one-to-one (each output has a unique input) in order for an inverse function to exist. Additionally, the function must be well-defined and continuous in order to have a meaningful inverse.

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