value of inverse function

Question

First day at university and some high school math, to "refresh". I need to find value $f^{-1}(12)$ of inverse function of $f(x)=5+x+\ln(x-6)$

Whatever I tried, I did not go far. Finally I found out about Lambert W function. We have never mentioned it at school, or I was sick that day :-P. This is what I did.

First subtract $11$ from both sides:

$$y=5+x+\ln(x-6)$$

$$y-11=x-6+\ln(x-6)$$

substitute:

$$x-6=Z$$

$$y-11=Z+\ln Z$$

exp:

$e^{y-11}=Z e^Z$ Lambert $W$ function:

$$Z=W(e^{y-11})$$

then I bring back subs:

$$x-6=W(e^{y-11})$$

$$x=6+W(e^{y-11})$$

Now, if I substitute y with 12, I get

$$x=6+W(e)$$

I also found a way how to calculate this in Python and got two values for $W(e)$: $1.76322281532$ and 1.76322284514

Which gave me results $x=7.76322281532$ and $x=7.76322284514$

At this point I wanted to verify the result and plugged x back into original function hoping to get y=12. And I got y=13.33.

I must have made a mistake but cannot find it.

Or I made more than one mistake?

Answer 1

You could also observe $f$ is monotonic, and try your luck (pardon, intuition) checking "easy" values. One value of the argument for which the logarithm is easy to calculate is $1$: you could then check $x = 7$ (as the argument of your logarithm is $x-6$) and verify $f(7) = 12$.

In reference to your approach, after finding the solution above I started to wonder if $W(e) = 1$ (I am not too conversant with the Lambert W function) and that indeed seems the case, https://en.wikipedia.org/wiki/Lambert_W_function, "Special Values" section, you might want to check your Python code.

Answer 2

Good hint for uni math is that questions will never have answers requiring maths above your grade level, which Lambert's W function definitely is.

Note that if $f(x)=y$, then $f^{-1}(y)=x$. So $f^{-1}(12)$ can just be found by setting $12=5+x+\ln(x-6)$, and you can take it from there.