test.html

<script src="mathjax/MathJax.js"> 
    MathJax.Hub.Config({ 
    extensions: ["mml2jax.js"], 
    jax: ["input/MathML", "output/HTML-CSS"] 
    }); 
</script>
<script>
(function () {
        var QUEUE = MathJax.Hub.queue;  // shorthand for the queue
        var math = null;                // the element jax for the math output.

        //
        //  Get the element jax when MathJax has produced it.
        //
        QUEUE.Push(function () {
          math = MathJax.Hub.getAllJax("MathOutput")[0];
          console.log('math',math);
        });

        //
        //  The onchange event handler that typesets the
        //  math entered by the user
        //
        window.UpdateMath = function (TeX) {
        	alert(TeX);
          QUEUE.Push(["Text",math,TeX]);
        }
      })();
</script>
<textarea onKeyUp="UpdateMath(this.value)" cols="100"></textarea>
<div id="MathOutput" >
You typed: ${}$
</div>