How to check whether a checkbox is checked in jQuery?

How do I successfully query the checked property?

The checked property of a checkbox DOM element will give you the checked state of the element.

Given your existing code, you could therefore do this:

if(document.getElementById('isAgeSelected').checked) {
    $("#txtAge").show();
} else {
    $("#txtAge").hide();
}

However, there's a much prettier way to do this, using toggle:

$('#isAgeSelected').click(function() {
    $("#txtAge").toggle(this.checked);
});
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<input type="checkbox" id="isAgeSelected"/>
<div id="txtAge" style="display:none">Age is something</div>