Please don't laugh at me for asking this question.
Why would you create a compound index with both last name and first name together? Why not two separate indexes where one is for last name and the other is for first name. That should handle any combination that you have in your where clause.
Because indexes don't work the way you think they do. Indexes sort off their key expression. If that expression contains two columns, the table is sorted by both. If it contains one column only, then its only sorted by that single column, period.
Without getting into too much technical detail, SQL contains an optimization called index intersection that allows two indexes to act a wee bit like one compound index. But "a wee bit" is the operative term here; the performance is not the same.
To use your firstname, lastname example. If you have a single "John Smith" in your table, a compound index will walk the tree right to that record-- all done! No scanning required. Two indexes, one on each column, will only allow the engine to find all Smiths, then find all Johns, then laboriously intersect the result result to find which ones match both criteria.
Performance takes a further hit when you want to order by multiple columns, as a single index will return the rows in proper order, whereas results from intersected indexes must be manually sorted each time.