Finding an inverse function

Question

I'd like to make sure I'm doing things right, my answer looks a little funny.

I have the following function:

$$g(x) = 3 + x + e^x$$

I am trying to find $g^{-1}(x)$, so I replace $g(x)$ with $y$ and switch it with $x$ to get:

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

I get y on a single side and end up with:

$$ g^{-1}(x) = -3-e^y + x $$

Is this correct? Should I have done something with logarithms to get rid of $-e^y$? How could I do it without also transforming the y on the other side?

Thanks!

Your method is incorrect; this function has no nice inverse. — vadim123, Jun 13 '14 at 22:56
According to wolfram alpha the function passes the horizontal line test. — Irresponsible Newb, Jun 13 '14 at 22:59
@Newb, quite right, it does have an inverse, just not a nice one. — vadim123, Jun 13 '14 at 23:05

MPW · Accepted Answer · 2014-06-14T01:16:12.307

2

This function is a 1-1 map of $\mathbb R$ onto itself, so it does have an inverse $g^{-1}$ that is also a map of $\mathbb R$ onto itself. Unfortunately, it is known that it is not possible to write this inverse explicitly in terms of elementary functions as you wish to do.

edited Jun 14 '14 at 01:16

answered Jun 13 '14 at 23:01

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Finding an inverse function

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