In modern browsers (IE8, Firefox, Chrome, Opera, Safari) you can use querySelector():
function test(el){
el.querySelector("#child").style.display = "none";
}
For older browsers (<=IE7), you would have to use some sort of library, such as Sizzle or a framework, such as jQuery, to work with selectors.
As mentioned, IDs are supposed to be unique within a document, so it’s easiest to just use document.getElementById("child").