for example:
- converting an X,Y,Z coordinate to Polar coordinate system
- or applying the Fourier transform and then inverse Fourier transform (or FFT)
- or in computer science, you give a utf8 string 'xyz' to encode then decode and get back the same value 'xyz'
Such transformations are said to be invertible, or to be bijections. If $T$ is the original transformation, then the transformation $S$ which "reverses" $T$ is called the inverse of $T$, and it's often denoted $T^{-1}$. The property that $S$ is an inverse of $T$ is expressed in the formal equations $$ TS = ST = \text{identity}, $$ which says in words that doing $S$ followed by $T$, or $T$ followed by $S$, is the same thing as doing the identity transformation.