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Assuming there's nothing wrong with the camera itself (possible), there are a few settings to check.

  • Hit the Menu button, go to the Setup menu (third icon), and select the Buttons item. There should be an option to set the behavior of the AF-L/AE-L button. When this is set to AF-ON, the button activates the focus, and the shutter release does not autofocus. (Some people prefer this behavior, called back-button focusback-button focus.) In any other setting, holding the shutter release down halfway will activate the autofocus, as you desire; pick whichever setting you prefer (you can hold the ?/zoom out button to see what each one does).
    That setting also affects live view if the focus mode is set to AF-S, but not if it's set to AF-F. In AF-F (available only in live view), the camera refocuses as the composition changes, not when you press the shutter release. I suspect you're using AF-F in live view.
  • I'm not sure about the problem you're having with the shutter release being disabled, but AFAIK not releasing the shutter when the camera can't focus is normal behavior. Internally, the viewfinder and live view use different methods to autofocus, which explains why the camera can focus in live view but not with the viewfinder. It's possible the scene you're trying to shoot is fooling the autofocus; the details of how autofocus works mean that it doesn't work with every scene (the D3200 manual good into more detail, I think). Without knowing your camera's settings or the types of scenes you're trying to shoot, I'm not sure what else to suggest. If you try a variety of scenes and it still won't work, it's possible you have a defective body. I'd try to find someone more knowledgeable who can test the camera before you try to get it repaired, though; a local camera shop may be able to help.

Assuming there's nothing wrong with the camera itself (possible), there are a few settings to check.

  • Hit the Menu button, go to the Setup menu (third icon), and select the Buttons item. There should be an option to set the behavior of the AF-L/AE-L button. When this is set to AF-ON, the button activates the focus, and the shutter release does not autofocus. (Some people prefer this behavior, called back-button focus.) In any other setting, holding the shutter release down halfway will activate the autofocus, as you desire; pick whichever setting you prefer (you can hold the ?/zoom out button to see what each one does).
    That setting also affects live view if the focus mode is set to AF-S, but not if it's set to AF-F. In AF-F (available only in live view), the camera refocuses as the composition changes, not when you press the shutter release. I suspect you're using AF-F in live view.
  • I'm not sure about the problem you're having with the shutter release being disabled, but AFAIK not releasing the shutter when the camera can't focus is normal behavior. Internally, the viewfinder and live view use different methods to autofocus, which explains why the camera can focus in live view but not with the viewfinder. It's possible the scene you're trying to shoot is fooling the autofocus; the details of how autofocus works mean that it doesn't work with every scene (the D3200 manual good into more detail, I think). Without knowing your camera's settings or the types of scenes you're trying to shoot, I'm not sure what else to suggest. If you try a variety of scenes and it still won't work, it's possible you have a defective body. I'd try to find someone more knowledgeable who can test the camera before you try to get it repaired, though; a local camera shop may be able to help.

Assuming there's nothing wrong with the camera itself (possible), there are a few settings to check.

  • Hit the Menu button, go to the Setup menu (third icon), and select the Buttons item. There should be an option to set the behavior of the AF-L/AE-L button. When this is set to AF-ON, the button activates the focus, and the shutter release does not autofocus. (Some people prefer this behavior, called back-button focus.) In any other setting, holding the shutter release down halfway will activate the autofocus, as you desire; pick whichever setting you prefer (you can hold the ?/zoom out button to see what each one does).
    That setting also affects live view if the focus mode is set to AF-S, but not if it's set to AF-F. In AF-F (available only in live view), the camera refocuses as the composition changes, not when you press the shutter release. I suspect you're using AF-F in live view.
  • I'm not sure about the problem you're having with the shutter release being disabled, but AFAIK not releasing the shutter when the camera can't focus is normal behavior. Internally, the viewfinder and live view use different methods to autofocus, which explains why the camera can focus in live view but not with the viewfinder. It's possible the scene you're trying to shoot is fooling the autofocus; the details of how autofocus works mean that it doesn't work with every scene (the D3200 manual good into more detail, I think). Without knowing your camera's settings or the types of scenes you're trying to shoot, I'm not sure what else to suggest. If you try a variety of scenes and it still won't work, it's possible you have a defective body. I'd try to find someone more knowledgeable who can test the camera before you try to get it repaired, though; a local camera shop may be able to help.
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Assuming there's nothing wrong with the camera itself (possible), there are a few settings to check.

  • Hit the Menu button, go to the Setup menu (third icon), and select the Buttons item. There should be an option to set the behavior of the AF-L/AE-L button. When this is set to AF-ON, the button activates the focus, and the shutter release does not autofocus. (Some people prefer this behavior, called back-button focus.) In any other setting, holding the shutter release down halfway will activate the autofocus, as you desire; pick whichever setting you prefer (you can hold the ?/zoom out button to see what each one does).
    That setting also affects live view if the focus mode is set to AF-S, but not if it's set to AF-F. In AF-F (available only in live view), the camera refocuses as the composition changes, not when you press the shutter release. I suspect you're using AF-F in live view.
  • I'm not sure about the problem you're having with the shutter release being disabled, but AFAIK not releasing the shutter when the camera can't focus is normal behavior. Internally, the viewfinder and live view use different methods to autofocus, which explains why the camera can focus in live view but not with the viewfinder. It's possible the scene you're trying to shoot is fooling the autofocus; the details of how autofocus works mean that it doesn't work with every scene (the D3200 manual good into more detail, I think). Without knowing your camera's settings or the types of scenes you're trying to shoot, I'm not sure what else to suggest. If you try a variety of scenes and it still won't work, it's possible you have a defective body. I'd try to find someone more knowledgeable who can test the camera before you try to get it repaired, though; a local camera shop may be able to help.