How can I test an AngularJS service from the console?

First of all, a modified version of your service.

a )

var app = angular.module('app',[]);

app.factory('ExampleService',function(){
    return {
        f1 : function(world){
            return 'Hello' + world;
        }
    };
});

This returns an object, nothing to new here.

Now the way to get this from the console is

b )

var $inj = angular.injector(['app']);
var serv = $inj.get('ExampleService');
serv.f1("World");

c )

One of the things you were doing there earlier was to assume that the app.factory returns you the function itself or a new'ed version of it. Which is not the case. In order to get a constructor you would either have to do

app.factory('ExampleService',function(){
        return function(){
            this.f1 = function(world){
                return 'Hello' + world;
            }
        };
    });

This returns an ExampleService constructor which you will next have to do a 'new' on.

Or alternatively,

app.service('ExampleService',function(){
            this.f1 = function(world){
                return 'Hello' + world;
            };
    });

This returns new ExampleService() on injection.

TLDR: In one line the command you are looking for:

angular.element(document.body).injector().get('serviceName')

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Deep dive

AngularJS uses Dependency Injection (DI) to inject services/factories into your components,directives and other services. So what you need to do to get a service is to get the injector of AngularJS first (the injector is responsible for wiring up all the dependencies and providing them to components).

To get the injector of your app you need to grab it from an element that angular is handling. For example if your app is registered on the body element you call injector = angular.element(document.body).injector()

From the retrieved injector you can then get whatever service you like with injector.get('ServiceName')

More information on that in this answer: Can't retrieve the injector from angular
And even more here: Call AngularJS from legacy code

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Another useful trick to get the $scope of a particular element. Select the element with the DOM inspection tool of your developer tools and then run the following line ($0 is always the selected element):
angular.element($0).scope()