
prüfen
to check
To check or examine something in order to determine whether it is correct, safe, or functioning properly.
Bevor wir losfahren, muss ich noch den Reifendruck prüfen.
Before we set off, I still need to check the tire pressure.
to check / test
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to check
To check or examine something in order to determine whether it is correct, safe, or functioning properly.
Bevor wir losfahren, muss ich noch den Reifendruck prüfen.
Before we set off, I still need to check the tire pressure.

to test
To determine a person's knowledge or abilities through an exam or test.
Der Professor wird uns nächste Woche im Fach Physik prüfen.
The professor will test us in physics next week.

to consider
To carefully consider and think through a possibility, proposal, or option before making a decision.
Wir prüfen derzeit, ob wir das Projekt finanzieren können.
We are currently considering whether we can finance the project.

to test
To put someone in a difficult challenge that tests their patience, character, or resilience.
Das Schicksal hat ihn mit vielen schweren Krankheiten hart geprüft.
Fate has tested him severely with many serious illnesses.
The verb 'prüfen' (to check / test) is the basis for the noun 'die Prüfung', which means 'test', 'exam', or 'inspection'. The person who checks or tests also has a name: 'der Prüfer' or 'die Prüferin' (examiner / inspector).
Die Studenten bereiten sich auf die mündliche Prüfung vor.
The students are preparing for the oral exam.
When you examine something for a specific quality or feature, you use the preposition 'auf' followed by the accusative. The structure is: 'etwas auf etwas prüfen' (to check something for something).
Die Werkstatt prüft das Auto auf mögliche Schäden.
The garage checks the car for possible damage.
When a person's knowledge in a particular subject is tested, you use the preposition 'in' with the dative. The structure is: 'jemanden in einem Fach prüfen' (to test someone in a subject).
Nächste Woche prüft sie uns in Geschichte.
Next week she will test us in history.
In the meaning of 'überdenken' (to rethink) or 'abwägen' (to weigh up), 'prüfen' (to consider) is very often used with a subordinate clause introduced by 'ob' (whether). This expresses an open possibility.
Die Firma prüft, ob eine Preiserhöhung notwendig ist.
The company is considering whether a price increase is necessary.
In the meaning of 'auf die Probe stellen' (to put to the test), 'prüfen' (to test) is often used in the passive. The subject is then the person experiencing the challenge.
Er wurde durch diese schwere Zeit hart geprüft.
He was severely tested by this difficult time.
to examine / investigate
Prüfen and untersuchen both mean looking at something closely. Prüfen usually means assessing whether something is correct, safe, or working properly. Untersuchen means systematically finding out new facts or a cause, especially in medicine and science.

Die Technikerin prüft den Motor vor der Fahrt auf seine Sicherheit.
The technician checks the engine for safety before the trip.
Die Technikerin untersucht den defekten Motor, um die Ursache des Fehlers zu finden.
The technician examines the defective engine to find the cause of the problem.