
die Aufgabe
task
A specific piece of work that has to be done, often as part of a larger activity, job, or education.
Meine wichtigste Aufgabe heute ist es, den Bericht fertigzustellen.
My most important task today is to finish the report.
task / function / surrender
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task
A specific piece of work that has to be done, often as part of a larger activity, job, or education.
Meine wichtigste Aufgabe heute ist es, den Bericht fertigzustellen.
My most important task today is to finish the report.

function
The function, purpose, or role that a person or thing has.
Es ist die Aufgabe des Herzens, Blut durch den Körper zu pumpen.
It is the function of the heart to pump blood through the body.

surrender / withdrawal
The ending or giving up of an activity, competition, or position, often because of difficulties or lack of success.
Nach einem Motorschaden zwang ihn die Technik zur Aufgabe des Rennens.
After engine damage, the technical problem forced him to withdraw from the race.

posting / mailing
The handing in of a postal item at a post office branch or parcel service.
Die Aufgabe des Einschreibens muss persönlich am Schalter erfolgen.
The posting of the registered letter must be done in person at the counter.
The noun 'die Aufgabe' (task/function/surrender) is derived directly from the verb 'aufgeben' (to give up / to assign / to mail). This connection explains the different meanings: you can 'aufgeben' a piece of work (assign it), 'aufgeben' a competition (give it up), or 'aufgeben' a letter (send it).
Der Trainer gab den Spielern die Aufgabe, ihre Pässe zu üben.
The coach gave the players the task of practicing their passes.
In the sense of work that needs to be done, 'Aufgabe' (task) is often combined with certain verbs. Typical phrases are: 'eine Aufgabe bekommen' (to get a task), 'eine Aufgabe erledigen' (to complete a task), 'eine Aufgabe lösen' (to solve a task/problem), or 'jemandem eine Aufgabe stellen' (to set someone a task).
Die Schüler müssen eine schwierige Aufgabe in Mathematik lösen.
The students have to solve a difficult math problem.
A very common construction is 'es ist die Aufgabe von jemandem, etwas zu tun' (it is someone's task to do something). This structure emphasizes the responsibility or function of a person or thing.
Es ist die Aufgabe der Regierung, für die Sicherheit der Bürger zu sorgen.
It is the government's task to ensure the safety of its citizens.
The meaning 'Beendigung' (ending/withdrawal) is almost always found in the context of sports, competitions, or fights. The surrender or withdrawal is often forced by external circumstances, e.g. by an injury or a technical defect.
Eine Verletzung am Knie zwang die Tennisspielerin zur Aufgabe des Spiels.
A knee injury forced the tennis player to withdraw from the match.
The meaning 'Abgabe einer Postsendung' (handing in/posting of a postal item) belongs to formal language, especially in official or business communication. It is often found in the terms and conditions of postal services.
Bitte bewahren Sie den Beleg über die Aufgabe des Pakets gut auf.
Please keep the receipt for the posting of the package in a safe place.
work
Arbeit and Aufgabe can refer to something that needs to be completed. However, an Aufgabe is usually a clearly defined unit or assignment, while Arbeit can also mean the entire work process and the effort required. Use Aufgabe for the specific assignment and Arbeit for the broader activity.

Die Aufgabe für heute ist, den Bericht zu prüfen.
Today’s task is to review the report.
Die Arbeit an dem Bericht dauert noch zwei Tage.
The work on the report will take two more days.