document.write( "Question 480142: 
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document.write( "Given f(x) = 3Öx-1, find f-1 if the inverse exists.\r\n" );
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Algebra.Com's Answer #328874 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Given f(x) = 3Öx-1 , if the inverse exist.\r\n" );
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document.write( "To do inverses you can't just learn the algebra part and skip the\r\n" );
document.write( "graph part because you have to understand what the inverse graph\r\n" );
document.write( "looks like because you have to know how to put in the proper\r\n" );
document.write( "restrictions.  So be sure you understand the fact that the inverse\r\n" );
document.write( "of a one-to-one function is its reflection across (or into) the\r\n" );
document.write( "identity line y=x which is a 45° line through the origin.  A graph\r\n" );
document.write( "and its inverse will always make a symmetrical figure about the\r\n" );
document.write( "identity line y=x.\r\n" );
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document.write( "Here is the graph of f(x) = 3Öx-1 \r\n" );
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document.write( "That 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 below 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) = 3Öx – 1\r\n" );
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document.write( "Then we change f(x) to y\r\n" );
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document.write( "     y = 3Öx – 1\r\n" );
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document.write( "Next we interchange x and y\r\n" );
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document.write( "     x = 3Öy – 1\r\n" );
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document.write( "We solve for y\r\n" );
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document.write( "Square both sides:    \r\n" );
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document.write( "  (x)² = (3Öy – 1)²\r\n" );
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document.write( "    x² = 9(y - 1)\r\n" );
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document.write( "    x² = 9y - 9\r\n" );
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document.write( "Swap sides:\r\n" );
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document.write( "9y - 9 = x²\r\n" );
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document.write( "    9y = x² + 9\r\n" );
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document.write( "Divide through by 9\r\n" );
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document.write( "     y = \"%28x%5E2%2B9%29%2F9\"\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) = \"%28x%5E2%2B9%29%2F9\"\r\n" );
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document.write( "By making two fractions and simplifying, \r\n" );
document.write( "that can be written equivalently as\r\n" );
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document.write( "f-1(x) = \"x%5E2%2F9%2B1\"\r\n" );
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document.write( "Either is correct if when we graph it we get the green curve above.\r\n" );
document.write( "But we must graph it to check to see if it is the inverse or not.\r\n" );
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document.write( "That should be 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( "However when we graph it, we get this: \r\n" );
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document.write( "So we must cut off the left part of the green graph.\r\n" );
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document.write( "The way we do that is to restrict the graph to the right\r\n" );
document.write( "side only, by requiring that x be greater than or equal 0,\r\n" );
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document.write( "So the answer is\r\n" );
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document.write( "f-1(x) = \"%28x%5E2%2B9%29%2F9\", x≧0\r\n" );
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document.write( "or\r\n" );
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document.write( "f-1(x) = \"x%5E2%2F9%2B1\",  x≧0\r\n" );
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document.write( "That way you get only the right side of that green graph:\r\n" );
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document.write( "But you MUST have the restriction x≧0 out beside it because the\r\n" );
document.write( "whole green parabola is NOT the inverse but only the right half of it.\r\n" );
document.write( "And the only way you would know to make that restriction is to draw\r\n" );
document.write( "and understand the graphs.\r\n" );
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document.write( "Edwin
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