# study this function : g(x) = x - 1 - ln(x)

study this function :

g(x) = x - 1 - ln(x)

## Complete answer with full step-by-step explanations

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**xxxxxxxx**

xxxxx xxxx xxxxxxxxx

**xxxxxxxxx**

**xxx xxxxxxxxxxx**

xx xxxxx the xxxx xxxx the derivation of the sum of xxxxxxxxx xx sun of xxxxxxxxxxx of functions:

xxx we apply xxx xxxxxx

and xx have:

**2nd derivation:**

xx can xxx 2nd xxxxxxxxxx xx deriving xxx xxx derivation:

xxx we xxxxx the rules:

And we have:

**Domain:**

xxxx xx xxx xxxxxxxx xx xxxxxxx xxx each xxxx xx xxx

xxxxxxxx xx xxx xxxxxxx for xxxx xxxx x, but only xxx positive xxxx values xx xx

xxx the domain xx xxx final function xxx

**Asymptotes:**

**vertical asymptotes:**

The xxxxxxxxxx for xxxxxxxx xxxxxxxxxx xxx xxxxxx on the edge xx the xxxxxxxx on xxxxx the xxxxxxxx is xxxxxxxx xxxxx xxx xxxxxxxx is defined on interval x xxx only xxxxx in xxxxx xx have to search for vertical asymptote is 0 xxx xxxx xxxx the right (positive side). xxxxxx xx xx point in finding the xxxxxxxx asymptote in xxxxxxxxxx xx xx have to solve the xxxxxx

Applying xxx xxxxx xx xxxxxxx we xxxxx

Now, for xxx xxxxx , xx know xxxx xxx xxxxx of

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## here is your answer(solution with graph)

## See attachment.

The analysis is fairly simple if you remember the definition of a Lambert function (Product Log) (see http://en.wikipedia.org/wiki/Lambert_W_function ).