Showing that a map is a diffeomorphism

Question

The exercises I am trying to solve is the following:

For a constant $a \in R$, let $f:R^3 \rightarrow R^3$ be given by $$f(x,y,z)=(x-\frac{az}{\sqrt{1+z^2}},y+\frac{a}{\sqrt{1+z^2}},z)$$ Show that $f$ is a diffeomorphism.

I have managed to show that it is 1-1 an onto. But I am having a problem with showing the smoothness of $f$ and $f^{-1}$. Thanks in advance.

Answer 1

$$f^{-1}(x,y,z)=\left(x+\frac{az}{\sqrt{1+z^2}},y-\frac{a}{\sqrt{1+z^2}},z\right)$$

You can verify that $ff^{-1}=id$ and $f^{-1}f=id$.

To show $f^{-1}$ is smooth, use the same approach as for $f$ in my comment.