I like Luis' answer. A different approach, likely with a different query plan which may be more or less efficient depending on your data/indexes, etc is this:
SELECT a.id,
a.first_name,
a.last_name,
b.income,
b.department
FROM CUSTOMERS a
CROSS APPLY (SELECT TOP 1 b.income, b.department FROM department b WHERE b.id = a.id ORDER BY b.income DESC) AS b