Inverse Composition Rules

f(x)	sin(x)	cos(x)	tan(x)	x2	sinh(x)	exp(x)
g(x)	arcsin(x)	arccos(x)	arctan(x)	√(x)	arcsinh(x)	log(x)
Composition	f(g(x))	g(f(x))

Code

HTML

<form name="input">
<table style="background-color:background; padding:5px; margin:2px">
 <tr style="color:blue">
   <th>f(x)</th>
   <td>sin(x)</td>
   <td>cos(x)</td>
   <td>tan(x)</td>
   <td>x<sup>2</sup></td>
   <td>sinh(x)</td>
   <td>exp(x)</td>
 </tr>
 <tr>
   <td>&nbsp;</td>
   <td><input type="radio" name="f" value="sin" checked onChange="change(this)"></td>
   <td><input type="radio" name="f" value="cos"         onChange="change(this)"></td>
   <td><input type="radio" name="f" value="tan"         onChange="change(this)"></td>
   <td><input type="radio" name="f" value="square"      onChange="change(this)"></td>
   <td><input type="radio" name="f" value="sinh"        onChange="change(this)"></td>
   <td><input type="radio" name="f" value="exp"         onChange="change(this)"></td>
 </tr>
 <tr style="color:green">
   <th>g(x)</th>
   <td>arcsin(x)</td>
   <td>arccos(x)</td>
   <td>arctan(x)</td>
   <td>&radic;(x)</td>
   <td>arcsinh(x)</td>
   <td>log(x)</td>
 </tr>
 <tr>
   <td>&nbsp;</td>
   <td><input type="radio" name="g" value="asin" checked onChange="change(this)"></td>
   <td><input type="radio" name="g" value="acos"         onChange="change(this)"></td>
   <td><input type="radio" name="g" value="atan"         onChange="change(this)"></td>
   <td><input type="radio" name="g" value="sqrt"         onChange="change(this)"></td>
   <td><input type="radio" name="g" value="asinh"        onChange="change(this)"></td>
   <td><input type="radio" name="g" value="log"          onChange="change(this)"></td>
 </tr>
 <tr style="color:red">
   <th>Composition</th>
   <td>f(g(x))</td>
   <td>g(f(x))</td>
   <td>&nbsp;</td>
   <td>&nbsp;</td>
   <td>&nbsp;</td>
   <td>&nbsp;</td>
 </tr>
 <tr>
   <td>&nbsp;</td>
   <td><input type="radio" name="fg" value="fg" checked onChange="comp(this)"></td>
   <td><input type="radio" name="fg" value="gf"         onChange="comp(this)"></td>
   <td>&nbsp;</td>
   <td>&nbsp;</td>
   <td>&nbsp;</td>
   <td>&nbsp;</td>
 </tr>
</table>
</form>

JavaScript

var brd = JXG.JSXGraph.initBoard('box', {axis:true, boundingbox: [-3, 3, 3, -3]});
var f, g, fg;

var change = function(obj) {
 var t = obj.value; 
 if (t=='sin')           { f = function(x) { return Math.sin(x); } 
 } else if (t=='cos')    { f = function(x) { return Math.cos(x); }
 } else if (t=='tan')    { f = function(x) { return Math.tan(x); }
 } else if (t=='square') { f = function(x) { return x*x; }
 } else if (t=='sinh')   { f = function(x) { return brd.sinh(x); }
 } else if (t=='exp')    { f = function(x) { return Math.exp(x); }

 } else if (t=='asin')   { g = function(x) { return Math.asin(x); }
 } else if (t=='acos')   { g = function(x) { return Math.acos(x); }
 } else if (t=='atan')   { g = function(x) { return Math.atan(x); }
 } else if (t=='sqrt')   { g = function(x) { return Math.sqrt(x); }
 } else if (t=='asinh')  { g = function(x) { return Math.log(x+Math.sqrt(1+x*x)); }
 } else if (t=='log')    { g = function(x) { return Math.log(x); }
 }
 plotf.Y = f; plotf.updateCurve();
 plotg.Y = g; plotg.updateCurve();
 brd.update();
}

var comp = function(obj) {
 var t = obj.value; 
 if (t=='fg')           { fg = function(x) { return f(g(x)); } 
 } else if (t=='gf')    { fg = function(x) { return g(f(x)); }
 }
 plotfg.Y = fg; plotfg.updateCurve();
 brd.update();
}

f = function(x) { return Math.sin(x); }
g = function(x) { return Math.asin(x); }
fg = function(x) { return Math.sin(Math.asin(x)); }

plotf = brd.create('functiongraph',[f,-2,2],{dash:2, strokeColor:'blue',strokeWidth:3});
plotg = brd.create('functiongraph',[g,-2,2],{dash:3, strokeColor:'green',strokeWidth:3});
plotfg = brd.create('functiongraph',[fg,-2,2],{strokeColor:'red',strokeWidth:3});