For instance in the snippet below - how do I access the h1 element knowing the ID of parent element (header-inner div)?
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If you were to use jQuery as mentioned by some posters, you can get access to the element very easily like so (though technically this would return a collection of matching elements if there were more than one H1 descendant):
var element = $('#header-inner h1');
Using a library like JQuery makes things like this trivial compared to the normal ways as mentioned in other posts. Then once you have a reference to it in a jQuery object, you have even more functions available to easily manipulate its content and appearance.