Learn Portuguese (Portugal): A função que o funcionário tem é muito importante. - The role that the employee has is very important.

No, in Portuguese you must always keep que. While in English you can say 'The role the employee has', in Portuguese you must explicitly connect the phrases by saying 'The role that the employee has' (A função que...).

Is funcionário a false friend? It looks like 'functionary'.

Yes, it can be a bit of a false friend! Funcionário is the everyday word for 'employee', 'staff member', or 'clerk'. It comes from the idea of someone who performs a function (função), but we rarely translate it as 'functionary' in English.

Why is it é and not está before muito importante?

We use é (from the verb ser) because being important is treated as an inherent, defining quality of this specific role. If you used está (from the verb estar), it would imply the role is only temporarily important right now.

Why doesn't the adjective importante change to match the feminine noun a função?

Adjectives ending in -e, like importante or grande, are gender-neutral in Portuguese. They keep the exact same form for both masculine and feminine nouns. They only change when pluralized (becoming importantes).