I've answered this in the past, but it's good that you're asking these questions.
$rootScope exists, but it can be used for evil Scopes in Angular form a hierarchy, prototypally inheriting from a root scope at the top of the tree. Usually this can be ignored, since most views have a controller, and therefore a scope, of their own.
Non-isolated scopes are hierarchical, but most developers should be using directives that have isolated scopes. The very hierarchical nature of AngularJS's scope is the source of many bugs in angular apps. It's a problem I like to call scope bleeding where a scope property is modified magically somewhere in the DOM tree and you don't know why.
Angular's default behavior is to inherent scopes and this makes it tempting for one controller to update something managed by another controller, so on, and so on. This is how spaghetti connections between source code is created. Making it very difficult to maintain that code.
Occasionally there are pieces of data that you want to make global to the whole app. For these, you can inject $rootScope and set values on it like any other scope.
No that's not correct. AngularJS allows you to define things like constants, values, and services. These are things that can be injected into routes, controllers and directives. That is how you make things accessible globally to your app, and this how you do it if you want to make your controllers or directives testable. A unit test writer doesn't know what properties should be in the $rootScope that a directive or controller depends upon. They have to assume that the $rootScope has not mutated to provide a service or data.
Of course, global state sucks and you should use $rootScope sparingly, like you would (hopefully) use with global variables in any language.
The problem isn't $rootScope but what people are doing with it. Many apps add the current user, the auth tokens, and the session data into the rootScope. This ends up getting used heavily in templates (shows X if user logged in otherwise show Y). The problem is that the HTML doesn't communicate scope hierarchy. So when you see {{user.firstname + ' ' + user.lastname}} you have no idea where the variable user came from. The second problem is child scopes can shadow root properties. As in the previous example if a directive does this scope.user = 'bla bla bla'. It hasn't replaced the value on the rootScope. It's hidden it. Now you get some strange unexpected things in the templates, and you don't know why the variable user has changed.
Conversely, don't create a service whose only purpose in life is to store and return bits of data.
Angular's $cacheFactory and $templateCache are examples of services that exist only too store data. I think the author was trying to encourage the use of constants and values in Angular's modules, but that's not a good description to do that.
So My doubt is why $rootScope is not recommended for functions as a global function? Is there any performance issue?
The $rootScope is the only scope available during angular.config(..). It's during this time that the scope can be modified if this is the only time that you can do it. For example; you may need to inject an API key or Google anayltics variable before the app starts.
Functions on any scope are generally a bad idea. Mainly for the reason that everything in scopes is digested in expressions on the templates. Functions tent to hide heavy operations. It's impossible to tell how heavy a template is by reading the HTML when it calls a function. I've seen scope functions like getHeight() where the function itself performed 3 levels of nested loops. That function has to get called every time angular digests the watchers to see if it's changed. You should try to keep your templates as dry as possible.
