Mapping in R³
- The map \( f: R³ \rightarrow R³ \) is defined as \( f(x, y, z) = (x, y, z) \), which means it simply maps a point in 3-dimensional space to itself.
- The map \( g: R³ \rightarrow R³ \) is defined as \( g(x, y, z) = (x+1, y-1, z) \), which means it shifts the x-coordinate by 1 unit to the right and the y-coordinate by 1 unit downwards, while leaving the z-coordinate unchanged.

Combining Maps
- When we combine the maps \( f \) and \( g \) by applying \( g \) first and then \( f \), denoted as \( fg \), we get:
\[ fg(x, y, z) = f(g(x, y, z)) = f(x+1, y-1, z) = (x+1, y-1, z) \]
- So, the combined map \( fg \) shifts the point in 3-dimensional space 1 unit to the right along the x-axis and 1 unit downwards along the y-axis, while leaving the z-coordinate unchanged.

Conclusion
- In summary, the map \( fg: R³ \rightarrow R³ \) shifts a point in 3-dimensional space to the right by 1 unit along the x-axis and downwards by 1 unit along the y-axis, while keeping the z-coordinate constant. This combined map is a composition of the individual maps \( f \) and \( g \), showing how multiple transformations can be applied successively to a point in space.