Algebra.Com's Answer #328758 by Edwin McCravy(20056)![\"\"](\"/images/stars/stars-13.gif\")  
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Given f(x) = 9 – x² , x≧3 , find f-1 if the inverse exist.
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document.write( "I'll just do the first one.\r\n" );
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document.write( "Here is the graph of f(x) = 9 – x² without the restriction x≧3\r\n" );
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document.write( "As you see it does not pass the horizontal line test so its inverse,\r\n" );
document.write( "(in green) which is its reflection in the 45° line through the origin whose\r\n" );
document.write( "equation is y=x (the blue dotted line below) would not pass the vertical \r\n" );
document.write( "line test, and would not be a function, as you can see:\r\n" );
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document.write( "However with the restriction x≧0, the graph is only the\r\n" );
document.write( "right half of the whole curve (without the restriction x≧0):\r\n" );
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document.write( "Then the curve passes the horizontal line test, so that when we\r\n" );
document.write( "reflect it in the line y=x (blue dotted line), the green inverse\r\n" );
document.write( "graph passes the vertical line test and therefore is a function:\r\n" );
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document.write( "Now we need to find the equation of the green functional curve,\r\n" );
document.write( "which is the graph of the inverse of the f(x), which is denoted\r\n" );
document.write( "f-1(x). \r\n" );
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document.write( "To do this, we start with the original equation:\r\n" );
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document.write( " f(x) = 9 – x²\r\n" );
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document.write( "Then we change f(x) to y\r\n" );
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document.write( " y = 9 - x²\r\n" );
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document.write( "Next we interchange x and y\r\n" );
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document.write( " x = 9 - y²\r\n" );
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document.write( "We solve for y\r\n" );
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document.write( " y² = 9 - x\r\n" );
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document.write( "Using the principle of square roots:\r\n" );
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document.write( " y = ±Ö9 - x\r\n" );
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document.write( "But we use only the + square root since the green graph of\r\n" );
document.write( "the inverse is above the x-axis, so we have\r\n" );
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document.write( " _____\r\n" );
document.write( " y = Ö9 - x\r\n" );
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document.write( "and now we replace y by f-1\r\n" );
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document.write( "f-1(x) = Ö9 - x\r\n" );
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document.write( "That's the equation of the green curve above which is the \r\n" );
document.write( "inverse of f(x)\r\n" );
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document.write( "Edwin \n" );
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