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Please tell me how to find the solution for this.

$$y= 3 + x + e^{x}$$

The range or the domain is not given. I'm just asked to find the inverse function of this.

Toiya
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  • Perhaps you should begin by showing us what you have tried to do. –  Apr 21 '13 at 01:22
  • And the context of the problem. It looks like part of a bigger problem where there is likely a much better approach than your first instinct on how to attack it. –  Apr 21 '13 at 01:24
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    @Toiya: I am pretty sure there is no closed formula for the inverse function using elementary functions. Who asked you this? – Stefan Smith Apr 21 '13 at 01:59
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    do you want inverse itself or just some info about it like number of roots , solution with other functions etc. I agree with @StefanSmith – ABC Apr 21 '13 at 02:20
  • @Toiya : see the answer below, but note that to know that the range is $\mathbb{R}$ you need not just that $y' > 0$ but $y' > 1$. – Stefan Smith Apr 23 '13 at 00:48

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Since $y'=1+e^{x}>0$, then $y$ is monotonically increasing on all of $\mathbb{R}$. That's why they don't state a domain and range-because the inverse function is defined on all of $\mathbb{R}$. As others have alluded to, there isn't a solution in terms of elementary functions. However, the following Mathematica commands get the inverse function in terms of the Lambert W function: \begin{align*} &f[x\_]:=3+x+\mathrm{Exp}[x]; \\ &\mathrm{InverseFunction}[f] \end{align*} returns $f^{-1}(x) = -3-W(\exp(x-3))+x$. (Mathematica calls the W function "$\rm{ProductLog}$") I'm a bit surprised that any homework problem would require you to pull out the Lambert W function. In any case, you can use the fact that $z=W(z)e^{W(z)}$ to verify that this is indeed the inverse function.

I have no clue how one would have gotten this result without a computer algebra system.

user14717
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