'Different work based on their functions’ should the word 'functions' in this sentence be function because the word 'Different' describes here that the word 'function' is plural.
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The word 'their' is a plural possessive pronoun (in this case, adjective). You have mentioned in the comments the antecedent of 'their' in the sentence is 'work', a singular noun. In English, possessive adjectives must agree in number with their antecedent, but not necessarily with the noun they modify.
The proper phrase would be 'Different work based on its function(s)'. The word 'its' must be used because 'different' is singular; however, whether or not function should be plural is your decision, and depends on the specific context of the sentence.
But (in this a big note), the original sentence is confusing. If you must ask this question, it signifies the sentence is ambiguous. One of the suggested versions in the comment would be 'Different types of work based on its function/their functions.'
Again, whether you choose 'their functions' or 'its function' is up to you.