@burcinc-- Yes, an x-ray is needed to diagnose the injury. But if your symptoms do not match those of a fracture, the doctor may not see the need for an x-ray. A fracture causes immense pain, swelling and the inability to move. A sprain causes these symptoms as well, but to a much lesser degree. For example, if you don't have any swelling and just some pain, you clearly don't have a fracture, just a mild sprain.
An x-ray is best when the symptoms are severe and the doctor can't tell if it's a fracture or a very serious sprain. An x-ray won't show the sprain, but if there is not fracture, there is only one other possibility. An MRI would be best to diagnose a sprain, but it's not usually preferred.