Test AngularJS factory function with Jasmine

I am very new to this (angularjs, jasmine, testacular) and I have this code (I simplified it a bit, leaving only what matters):

//my_module.js
angular.module('my_module', ['my_data'])
.config([...]);

.controller('my_controller', ['$scope', 'my_data',
    function($scope, my_data) {
        $scope.my_function = function() {
            return my_data.my_factory.save().then(function () {
                console.log('saved');
            },
            function() {                             
                console.log('Error');
            }); 
        }
    }
)

//my_data.js
angular.module('my_data', [])
.factory('my_factory', ['$q', '$rootScope',
    function($q, $rootScope) {
        var my_factory= function(my_data) {
            angular.extend(this, my_data);
        }
        my_factory.prototype.save = function() {
            var deferred = $q.defer();
            setTimeout(function() {
                deferred.resolve();
            $rootScope.$apply();
            }, 1000);

            return deferred.promise;
        }
        return my_factory;
    }
])

So, what I want to do is to test if my_data.my_factory.save is called when the my_module.my_controller.my_function is fired.

//my_test.js
describe('testing my_controller.my_function', function () {
    beforeEach(module('my_module'));

    var $rootScope, $controller;
    beforeEach(inject(function(_$rootScope_, _$controller_) {
        $rootScope = _$rootScope_;
        $controller = _$controller_;
    }));

    scope = $rootScope.$new();

    it('should call the save function', function() {
        scope.my_function();
        expect(my_data.save).toHaveBeenCalled();
    });
}

I need a little bit of help.

You're not too far from what you need. First off, as you require my_data as my_module dependency, you don't need to inject my_module to the controller, just the factory (my_factory);

Secondly, you want to make use of ngMock. The docs are not very complete, but give a good insight. More here and a example here (look for test/unit/controllers).

Basically, what you want to do is to mock the service so you can be assured it has been called. To achieve it, inject $provide to your angular.mock.module call and provide a mocked my_factory service. The best way to achieve it is something like this:

describe('testing my_controller.my_function', function () {
  var mockedFactory, $rootScope, $controller;

  beforeEach(module('my_module', function($provide) {
    mockedFactory = {
      save: jasmine.createSpy()
    };

    $provide.value('my_factory', mockedFactory);
  }));

  beforeEach(inject(function(_$rootScope_, _$controller_) {
    $rootScope = _$rootScope_;
    $controller = _$controller_;
  }));

  scope = $rootScope.$new();

  it('should call the save function', function() {
    scope.my_function();
    expect(mockedFactory.save).toHaveBeenCalled();
  });
}

This way you'll override my_factory dependency.