In my math assignment I have to find the inverse of
$$ f(x_1, x_2) = \left(\ln \left(\frac{x_2}{x_1}\right), x_1^2 + x_2^2\right) $$
Now I already have looked into this, and came up with the following:
I split f into $$ f_1(x_1,x_2) = \ln \left(\frac{x_2}{x_1} \right)\\ \text{and} \\ f_2(x_1,x_2) = x_1^2+x_2^2 $$
solving $f_1$ for $x_2$ resulted in $$e^{f_1} \cdot x_1 = x_2 $$
and after using this $x_2$ in $f_2$ $$ x_1 = \sqrt{\frac{f_2}{1+e^{2f_1}}} $$
which I then used in $x_2$ again and led to:
$$ x_2 = e^{f_1} \cdot \sqrt{\frac {f_2}{1+e^{2f_1}}} $$
So according to what I have done the inverse should be:
$$ f^{-1}(f_1, f_2) = \left( \sqrt{\frac{f_2}{1+e^{2f_1}}} \ ,\ e^{f_1} \cdot \sqrt{\frac {f_2}{1+e^{2f_1}}} \right) $$
But after looking at another example on this site, it appears to be wrong. So I'd like to ask whether this is wrong or not, and if it is where I made the mistakes.
$$ f^{-1}(f_1, f_2) = \left( \sqrt{\frac{f_2}{1+e^{2f_1}}} \ ,\ e^{f_1} \cdot \sqrt{\frac {f_2}{1+e^{2f_1}}} \right) $$
was indeed correct.