aufgreifen
to take up
To notice an idea, thought, topic, or suggestion and pursue it further or respond to it.
Ich möchte den Punkt aufgreifen, den Sie vorhin erwähnt haben.
I would like to take up the point you mentioned earlier.

to take up / pick up
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to take up
To notice an idea, thought, topic, or suggestion and pursue it further or respond to it.
Ich möchte den Punkt aufgreifen, den Sie vorhin erwähnt haben.
I would like to take up the point you mentioned earlier.

'Aufgreifen' (to take up/pick up) is a separable verb. The prefix 'auf-' stands at the end in a main clause. In a subordinate clause, the verb stays together, and in the infinitive with 'zu', 'zu' goes in the middle.
Der Journalist griff das Thema in seinem Artikel auf.
The journalist took up the topic in his article.
The verb consists of the prefix 'auf-' and the verb 'greifen' (to grab). The literal meaning, 'to reach up and take,' helps you understand the abstract meanings too: you 'take up' an idea or a person.
Können wir diesen Punkt später noch einmal aufgreifen?
Can we take up this point again later?
In the abstract meaning, 'aufgreifen' (to take up) is often combined with words from the area of communication and ideas. Typical phrases are: ein Thema aufgreifen (to take up a topic), eine Frage aufgreifen (to take up a question), einen Gedanken aufgreifen (to take up a thought), or eine Anregung aufgreifen (to take up a suggestion).
Die Rednerin griff eine Frage aus dem Publikum auf.
The speaker took up a question from the audience.
When 'aufgreifen' (to pick up) has a person as its object, it almost always refers to a police or official context. It then means to find a person and take them along or arrest them.
Ein Spaziergänger fand das vermisste Kind und die Polizei griff es auf.
A walker found the missing child, and the police picked the child up.
to record / take in
'aufnehmen' and 'aufgreifen' can both express that you continue with something or deal with it again. 'aufnehmen' is used more for an activity, work, or task that begins or continues after a break. 'aufgreifen' is used more for an idea, suggestion, or topic that someone notices and then follows up on. The words can be confused because both occur in connection with work or conversations. More natural is: You 'nimmt die Arbeit wieder auf' (resume work), but you 'greift einen Vorschlag oder ein Thema auf' (take up a suggestion or topic).

Im Unterricht greift Julia die Frage ihres Freundes auf.
In class, Julia takes up her friend's question.
Nach der kurzen Pause nimmt sie die Arbeit am Bericht wieder auf.
After the short break, she resumes work on the report.