We define a mapping $f$ of a topological space $X$ into a topological space $Y$ to be proper if the subspace $f^{-1}(C)$ is compact in $X$ whenever $C$ is a compact subspace of $Y$.
Now how to determine if the self-maps $f$, $g$ of the reals defined as $f(x):= x^m$ and $g(x):= a - x^m$ for all real $x$, where $a$ is real and $m$ a positive integer, are proper or not?
First recall that the image of a compact set under a continuous function between topological spaces is itself compact. This is proved by taking the given open cover on the image, using continuity to get an open cover of the original set, using compactness to reduce this to a finite open cover, and then taking the corresponding finite cover of the image set.
Then notice how if $m$ is odd, then $f$ and $g$ have (continuous) inverses $f^{-1}(y) = ~ ^m \sqrt y$ and $g^{-1}(y) =~^{m} \sqrt {a -y}$ which must map compact sets to compact sets.
If $m$ is even the problem is slightly harder. If $C$ is compact then the inverse map to $f$ is only defined over the positive numbers. So $f^{-1}(C)= f^{-1}(C ~\cap~[0, \infty) )$ Since $C$ is compact in $ \mathbb{R}$ it is also closed. $[0, \infty)$ is also closed, so the intersection is a closed subset of a compact set, which requires it to be compact. The inverse function to $f$ is well defined over our new compact set and thus maps it to a compact set.
The even case for $g$ can be done by a similar method. The only difference is that restriction of domain we must make to define the inverse map is a bit more complicated.