document.write( "Question 1100913: Determine the values of the variable for which the expression is defined as a real number.\r
\n" ); document.write( "\n" ); document.write( "fourth root of(1-x)/(2-x) \n" ); document.write( "
Algebra.Com's Answer #715488 by ikleyn(52786)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "    Find the domain of the function  fourth root of (1-x)/(2-x).\r\n" );
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document.write( "f(x) = \"root%284%2C%281-x%29%2F%282-x%29%29\".      (1)\r\n" );
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document.write( "The domain is the set of those real numbers, where the expression under the root is non-negative:\r\n" );
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document.write( "    \"%281-x%29%2F%282-x%29\" >= 0.      (2)\r\n" );
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document.write( "So, we need to find the solution set of this inequality.\r\n" );
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document.write( "The rational function (2) has two critical points: x= 1  and  x= 2  (in ascending order).\r\n" );
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document.write( "1)  If x <= 1, then both binomial are non-positive:  \r\n" );
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document.write( "              1-x <= 0  and  2-x < 0.\r\n" );
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document.write( "              Hence, the ratio  \"%281-x%29%2F%282-x%29\" is non-negative in this domain \r\n" );
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document.write( "              and the set  x <= 1  DOES BELONG  to the set of solutions to inequality (2).\r\n" );
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document.write( "2)  If 1 < x <2, then the binomial 1-x is positive  and  the binomial 2-x is negative:\r\n" );
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document.write( "              1-x > 0   and  2-x < 0.\r\n" );
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document.write( "              Hence, the ratio  \"%281-x%29%2F%282-x%29\" is negative in this domain \r\n" );
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document.write( "              and the set  1 < x < 2  DOES NOT BELONG  to the set of solutions to equation (2).\r\n" );
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document.write( "2)  If x > 2, then both binomials  1-x  and  2-x  are greater than 0:  \r\n" );
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document.write( "              1-x > 0  and  2-x > 0.\r\n" );
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document.write( "              Hence, the ratio  \"%281-x%29%2F%282-x%29\" is positive in this domain \r\n" );
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document.write( "              and the set  x > 1  DOES BELONG  to the set of solutions of inequality (2).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answer. The domain of the given function is the set (\"-infinity\",1] U (2,\"infinity\").\r
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\n" ); document.write( "To see more examples of solved problems on inequalities and finding function domains, look into these relevant lessons\r
\n" ); document.write( "\n" ); document.write( "    - Solving problems on quadratic inequalities,\r
\n" ); document.write( "\n" ); document.write( "    - Solving inequalities for high degree polynomials factored into a product of linear binomials \r
\n" ); document.write( "\n" ); document.write( "    - Solving inequalities for rational functions with numerator and denominator factored into a product of linear binomials \r
\n" ); document.write( "\n" ); document.write( "    - Solving inequalities for rational functions with non-zero right side \r
\n" ); document.write( "\n" ); document.write( "    - Advanced lesson on inequalities\r
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\n" ); document.write( "\n" ); document.write( "    - Domain of a function which involves a quadratic polynomial under the square root operator,\r
\n" ); document.write( "\n" ); document.write( "    - Domain of a function containing high degree polynomial under the square root operator \r
\n" ); document.write( "\n" ); document.write( "    - Domain of a function which is the square root of a rational function\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lesson is the part of this online textbook under the topic  \"Inequalities\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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