You need to check if it has class, not if it is, as you have several on times. As you use jQuery you can use .hasClass(), .addClass() and removeCLass(). You might also want to look at .toggleClass().
function changeArrow(element) {
var $arrow = $(element);
if ($arrow.hasClass('background_one')) {
$arrow.removeClass('background_one');
$arrow.addClass('background_two');
} else {
$arrow.removeClass('background_two');
$arrow.addClass('background_one');
}
}