【Fasten】 , 【fix】 ,  【attach】 ,  【affix】  mean to make something stay firmly in place or in an assigned place. All but  【fix】  (and that sometimes) imply a uniting or joining of one thing to another or of two things together.

【Fasten】 implies an attempt to keep a thing from moving by uniting it (as by tying, binding, nailing, or cementing) to something else or by restraining it by means of some mechanical device (as a lock, a screw, or a hook and eye).

【Fix】 implies an attempt to keep something from falling down or from losing hold; it suggests such operations as driving in or implanting deeply, usually with care and accuracy.

It is more common in its extended than in its basic sense, but the implications remain the same. In some phrases where  【fasten】  and  【fix】  are used interchangeably there may be a distinction in meaning which is subtle but justified; thus, to  【fix】  one's affections on someone connotes concentration and fidelity while to  【fasten】  one's affections on someone may, and often does, suggest covetousness or an attempt to hold or control; to  【fix】  the blame upon a person implies solid grounds for the accusation, but to  【fasten】 the blame upon someone often suggests factitious grounds or selfish motives.

【Attach】 stresses connection or union in order to keep things together or to prevent their separation; it usually implies a bond, link, or tie.

【Affix】 usually implies imposition of one thing upon another; it may convey no further information, but it more often than not suggests either attachment by an adhesive (as paste, gum, or mucilage) or subscription (as of a name to a document).