Functions
A function is a piece of code that is called by name. It can be passed data to operate on (ie. the parameters) and can optionally return data (the return value).All data that is passed to a function is explicitly passed.
Let's say a function is a block of code (usually with its own scope, and sometimes with its own closure) that may receive some arguments and may also return a result.
Methods
A method is a piece of code that is called by name that is associated with an object. In most respects it is identical to a function except for two key differences.1. It is implicitly passed the object for which it was called
2. It is able to operate on data that is contained within the class (remembering that an object is an instance of a class - the class is the definition, the object is an instance of that data)
A method is a function that is owned by an object (in some object oriented systems, it is more correct to say it is owned by a class). Being "owned" by a object/class means that you refer to the method through the object/class; for example, in Java if you want to invoke a method "open()" owned by an object "door" you need to write "door.open()".
The way objects have to expose their behavior are called methods. Methods thus are the artifact object have to "do" something.
In many object oriented languages, all "functions" belong to some object (or class) and so in these languages there are no functions that are not methods.
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