Long Problem (Question 2)

Solution:

2a) $ f(g(x)) = x $ if $ x \ge 0 $, otherwise $ f(g(x)) $ is undefined assuming the range of f and g are also real. If the ranges are complex then the $ f(g(x)) = x $ for all real $ x $.

2b) $ g(f(x)) = |x| $ for all $ x \in \mathbf R $.
2c) $ g(f(-x)) = |x| $ for all $ x \in \mathbf R $.
2d) No, they are not, even if the range is complex.