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.