document.write( "Question 1209614: Suppose the domain of f is (-1,3). Define the function g by
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Algebra.Com's Answer #849666 by ikleyn(52794) $\"\"$ $\"About$

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\n" ); document.write( "Suppose the domain of f is (-1,3). Define the function g by
\n" ); document.write( "g(x) = 5 - f(x) + f(5/x).
\n" ); document.write( "What is the domain of g?
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document.write( "Let D be the domain of f(x),  D = (-1,3).\r\n" );
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document.write( "Both x and 5/x should be in the interval (-1,3),\r\n" );
document.write( "and, additionally, x should not be equal to 0 (zero).\r\n" );
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document.write( "So, the constraints are\r\n" );
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document.write( "    -1 < x < 3,    (1)\r\n" );
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document.write( "    -1 < 5/x < 3,  (2)\r\n" );
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document.write( "     x =/= 0.      (3)\r\n" );
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document.write( "Inequalities (1) and (3) tell us that we should consider two separate intervals (-1,0) and (0,3) for x,\r\n" );
document.write( "and determine other limitations on x from inequality (2) \r\n" );
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document.write( "(a)  So, let x be in the interval (-1,0).  Thus, x is negative now.\r\n" );
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document.write( "     Then first of the two inequalities (2),  -1 < 5/x, is equivalent to\r\n" );
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document.write( "         -x > 5  (after multiplying both sides by negative value of x and flipping the inequality sign),\r\n" );
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document.write( "     or, which is the same,\r\n" );
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document.write( "          x < -5.\r\n" );
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document.write( "          Thus we determined that if x is in (-1,0), then due to first inequality (2), \r\n" );
document.write( "                  x must be lesser than -5, which is out of the domain D.\r\n" );
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document.write( "          So, we may exclude this case \"x is in (-1,0)\"  from our consideration.\r\n" );
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document.write( "(b)  Now, let x be in the interval (0,3).  Thus, x is positive now.\r\n" );
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document.write( "     Then first of the two inequalities (2),  -1 < 5/x, is always valid and does not imply other restrictions on x.\r\n" );
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document.write( "     The second of the two inequalities (2),  5/x < 3,  then implies  x > 5/3.\r\n" );
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document.write( "            Thus, if x is positive, then due to second inequality of (2), \r\n" );
document.write( "                        x must be greater than 5/3.\r\n" );
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document.write( "Combining what we found in (a) and (b), the answer to the problem's question is\r\n" );
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document.write( "     +-------------------------------------------------+\r\n" );
document.write( "     |   The domain of the function g(x) is (5/3,3).   |\r\n" );
document.write( "     +-------------------------------------------------+\r\n" );
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