Gender is not a problem when it comes to cheering on my favorite Taiwanese baseball team, the Brother Elephants.


Could I replace 'cheering on" in the above with "cheering for//rooting for?" Thanks.
Rooting is more casual. Cheering on suggests more specific encouragement to win than does cheering for.
cheer on is a (separable) phrasal verb.
to cheer on the team; to cheer the team on; to cheer them on.
cheer for is a normal verb with a preposition.
to cheer for the team.
on has the sense of encouragement of continuing forward motion.
to cheer on; to urge on; to egg on.

for has the sense of being in favor of someone or something.
to cheer for; to vote for; to stand up for.
(Leaving out on sounds weird to my ear.)
Students: Are you brave enough to let our tutors analyse your pronunciation?
Thanks, Mister.

By the way, could I omit 'on' after "cheering?" If not, what does "on" refer to?
 CalifJim's reply was promoted to an answer.
Students: We have free audio pronunciation exercises.
Thanks, Jim.

Got it.