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How are they different? Does the word 'hurt' in no. 1 function as an adjective?

1. Are you hurt?
2. Are you hurting? (this seems to impIy present-time pertinence, IMO)
3. Does it hurt? (this seems to be different than the ones above in terms of meaning)
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Hi,

How are they different? Does the word 'hurt' in no. 1 function as an adjective? Yes

1. Are you hurt? Similar to 'Do you have an injury?
2. Are you hurting? (this seems to impIy present-time pertinence, IMO) Yes.
3. Does it hurt? (this seems to be different than the ones above in terms of meaning) This refers more specifically to the injured place, or to the cause of the injury.
eg I see you have a big cut on your leg. Does it hurt?
eg Does it hurt when I stick this knife in your leg?
It can also refer to problems that do not originate from outside your body.
eg I know you have arthritis in your knee. Does it hurt?

'Hurt' can also be used in this manner.
eg Stop squeezing my arm. You are hurting me.
eg When he is angry, he hurts people.

Best wishes, Clive
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"Are you hurting?" (Are you in pain?) is probably the most likely of the three to be used figuratively: "Are you hurting for [lack of] money?" The others tend to refer to "anguish" when not referring to physical pain.

I agree, #3 refers to the source of the pain rather than the person's condition.
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No. 'I was hurt by someone'.