document.write( "Question 908997: For the function f(x) = (1-x)/x , use composition of functions to show that f –^1 (x) = 1/(x+1)
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Algebra.Com's Answer #551466 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The red graph below is the graph of \"%22f%28x%29%22+=++%281-x%29%2Fx\".\r\n" );
document.write( "The green dotted line is the graph of the identity function \"%22I%28x%29%22+=+x\"\r\n" );
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document.write( "The inverse function f-1 is the reflection of the graph\r\n" );
document.write( "of f(x) in or across the identity line whose equation is I(x)=x, which\r\n" );
document.write( "is the green dotted line above.\r\n" );
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document.write( "Let's draw the graph of  f-1 = \"1%2F%28x%2B1%29\" (in blue) to see it it looks like\r\n" );
document.write( "it really it is the reflections of the red graph f(x) in or across the identity\r\n" );
document.write( "function, which is the green dotted line I(x)=x:\r\n" );
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document.write( "Yes it does look like it is.  To show this algebraically we must find the\r\n" );
document.write( "composition f∘-1 and also the composition  f-1∘f(x) and show that in both cases \r\n" );
document.write( "we get the right side of the identity function I(x)=x, which is x.\r\n" );
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document.write( "f∘f-1(x) = \"%281-%281%2F%28x%2B1%29%29%29%2F%281%2F%28x%2B1%29%29\"\r\n" );
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document.write( "Multiply top and bottom by LCD of (x+1)\r\n" );
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document.write( "f∘f-1(x) = \"%281%28x%2B1%29-1%29%2F1\"\r\n" );
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document.write( "f∘f-1(x) = \"%28x%2B1-1%29%2F1\"\r\n" );
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document.write( "f∘f-1(x) = \"x\"\r\n" );
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document.write( "So it gives x, which is the right side of the identity function I(x)=x,\r\n" );
document.write( "which is the green dotted line.\r\n" );
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document.write( "But we also have to show that it refects into I(x)=x both ways.\r\n" );
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document.write( "f-1∘f(x) = \"1%2F%28%281-x%29%2Fx%2B1%29\"\r\n" );
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document.write( "Multiply top and bottom by LCD of x\r\n" );
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document.write( "f-1∘f(x) = \"x%2F%281-x%2Bx%29\"\r\n" );
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document.write( "f-1∘f(x) = \"x%2F1\"\r\n" );
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document.write( "f-1∘f(x) = \"x\"\r\n" );
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document.write( "We have found that both f∘f-1(x) and f-1∘f(x) are equal to the right side \r\n" );
document.write( "of the identity function I(x) = x, and that proves that each is the reflection\r\n" );
document.write( "of the other in the green dotted line which is the identity function I(x) = x,\r\n" );
document.write( "because composition with each other gives the right side of I(x) = x, which is\r\n" );
document.write( "just x.\r\n" );
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document.write( "Edwin
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