# Difference between revisions of "Inverse Composition Rules"

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

### The underlying JavaScript code

<html>
<form name="input">
<table>
<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>
</tr>
<tr style="color:green">
<th>g(x)</th>
<td>arcsin(x)</td>
<td>arccos(x)</td>
<td>arctan(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>
</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>&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, strokeColor:'blue',strokeWidth:3});
plotg = brd.createElement('functiongraph',[g,-2,2],{dash:3, strokeColor:'green',strokeWidth:3});
plotfg = brd.createElement('functiongraph',[fg,-2,2],{strokeColor:'red',strokeWidth:3});
</jsxgraph>