How can I add a prefix to titles displayed in sidebar using function.php?

you can filter widget titles by adding below filter :

add_filter ('widget_title', 'custom_title');
function custom_title($oldtitle) 
{
    //manipulate $title as per your need
    $title = explode(" ", $oldtitle,2);
    $titleNew = "<span>".$title[0]."</span>."$title[1];
    return $titleNew;
}