Difference between revisions of "Inverse Composition Rules"

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===The underlying JavaScript code===
 
===The underlying JavaScript code===
 
<source lang="xml">
 
<source lang="xml">
 +
<html>
 +
<form name="input">
 +
<table>
 +
<tr>
 +
  <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>
 +
  <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>
 +
  <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>
 +
</html>
 +
<jsxgraph width="600" height="600" box="box">
 +
var brd = JXG.JSXGraph.initBoard('box', {axis:true, originX: 300, originY: 300, grid:true, unitX: 100, unitY: 100});
 +
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.createElement('functiongraph',[f,-2,2],{dash:2});
 +
plotg = brd.createElement('functiongraph',[g,-2,2],{dash:3, strokeColor:'black'});
 +
plotfg = brd.createElement('functiongraph',[fg,-2,2],{strokeColor:'red'});
 +
 +
</jsxgraph>
 
</source>
 
</source>
  
 
[[Category:Examples]]
 
[[Category:Examples]]
 
[[Category:Calculus]]
 
[[Category:Calculus]]

Revision as of 15:47, 19 June 2009

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)        
         

The underlying JavaScript code

<html>
<form name="input">
<table>
 <tr>
   <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>
   <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>
   <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>
</html>
<jsxgraph width="600" height="600" box="box">
var brd = JXG.JSXGraph.initBoard('box', {axis:true, originX: 300, originY: 300, grid:true, unitX: 100, unitY: 100});
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.createElement('functiongraph',[f,-2,2],{dash:2});
plotg = brd.createElement('functiongraph',[g,-2,2],{dash:3, strokeColor:'black'});
plotfg = brd.createElement('functiongraph',[fg,-2,2],{strokeColor:'red'});

</jsxgraph>