Just for wizards!!!

[stefano]

Hi there,

If I want to find the inverse function of:

"Y = X + 4"

I think to find it immediatly (X = Y - 4)



But for the following function:

Y = X + e^x

with "e" = napier (2.7 ecc.)


What is the inverse function?

[parksie]

As a quick FYI, Naperian logs aren't necessarily natural logs

I don't have any idea about the solution though

[Good Dreams]

How do we calculate e?

[parksie]

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + ...

[stefano]

Wait a moment,

I need to transform y=f(x) to x=f(y) of the following function:

y = x + e^x

okay?

[Guv]

In theory, there is an inverse function because the function is continuous and differentiable. From a more intuitive point of view, you can plot Y versus X and see that it is a smooth curve. If you turn the plot on its side, it is a plot of X versus Y.

While an inverse function exists in theory, it might not be possible to come up with the analytical expression for it. From a practical point of view, you can almost always come up with numerical solutions to such problems. It is analogous to finding roots for 5th (and higher) order polynomials. Theoretically they exist, and numerically you can always find them to any precision you desire. Except for special cases, it has been proven that there is no analytical/algebraic expression for the roots of polynomials beyond the 4th order.

Do you have reason to believe that there is some simple inverse function? If so, the trickery required to determine it seems obscure.

As X grows without an upper bound, e^X is a limiting function. Id est: (e^X + X)/e^X approaches one for increasing positive values. For example.

• (e^20 + 20)/e^20 = 1.000000041

Therefore for extreme positive values, X = ln(Y) is a good approximation.

As X decreases without a lower bound, X is the limiting function. Id est: (e^X + X)/X approaches one for decreasing negative values. For example.

• (e^-10 - 10)/-10 = .99999546

Therefore for extreme negative values, X = Y is a good approximation.

For every value of Y, you can use the Newton-Raphson (? Spelling) to solve for X. For Y = Constant use Newton-Raphson method to find a value for which F(X) = zero, where

• F(X) = E^X + X - Constant

Perhaps you could work with the series representation of the function.

• Y = 1 + 2*X/1! + X^2/2! + X^3/3! + X^4/4! + X^5/5! . . .

Some math text might have a way of inverting the above series. If so, I do not know what it is.

You could use the above to come up with a “Piecewise” approximating function.

• Use X = ln(Y) for large positive values of Y
• Use X = Y for extreme negative values.
• Use a least squares polynomial approximation for intermediate values.

If there is an analytical inverse, I would like to know what it is and how to derive it.

[stefano]

Thank you very much. Since I wrote the question you are the one that answered (in part) to it!