document.write( "Question 1208035: Find the domain, range, and intercepts of the function\r
\n" ); document.write( "\n" ); document.write( "A(x) = 4x•sqrt{1 - x^2}. Finally, graph the function. \r
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Algebra.Com's Answer #846166 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Find the domain, range, and intercepts of the function
\n" ); document.write( "A(x) = 4x•sqrt{1 - x^2}. Finally, graph the function.
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\n" ); document.write( "\n" ); document.write( "        In this problem, finding domain is a simple elementary task: the domain is [-1,1].\r
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\n" ); document.write( "\n" ); document.write( "        Finding x-intercepts is also simple elementary task: they are -1, 0 and 1 on x-axis.\r
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\n" ); document.write( "\n" ); document.write( "        The key issue is to find the range.
\n" ); document.write( "        Tutor Edwin made it in his post, using derivatives and Calculus; it requires a painstaking technique.\r
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\n" ); document.write( "\n" ); document.write( "        Here I will show another way to find the range, which uses simple elementary Algebra with minimum calculations.\r
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document.write( "Let real number \"t\" belongs to the range.  It means that\r\n" );
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document.write( "     = t    (1)\r\n" );
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document.write( "for some value of x.  Square bot sides of (1).  You will get\r\n" );
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document.write( "     = t^2\r\n" );
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document.write( "     = 0.    (2)\r\n" );
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document.write( "You can consider equation (2) as a quadratic equation relative .\r\n" );
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document.write( "The condition that it has a real solution for   is this inequality for the discriminant\r\n" );
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document.write( "     >= 0,  or   >= 0,  or  256 >= .\r\n" );
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document.write( "It implies   <= 4;  hence,  |t| <= 2,  or  -2 <= t <= 2.\r\n" );
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document.write( "Thus, equation (1) has a real solution if and only if  -2 <= t <= 2.\r\n" );
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document.write( "So, the range of the function    is this interval  -2 <= t <= 2,  or t belongs  [-2,2].    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "Thus, this problem can be solved in this simple way much easier than applying Calculus.\r
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\n" ); document.write( "\n" ); document.write( "Surely, the Calculus approach is like a heavy army tank: it is universal and works everywhere.\r
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\n" ); document.write( "\n" ); document.write( "So, if you know Calculus and do not afraid to apply it, boldly go forward.\r
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\n" ); document.write( "\n" ); document.write( "But if the teacher gave you similar problem long before you learn Calculus,
\n" ); document.write( "know and remember that this method from my post probably works.\r
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\n" ); document.write( "\n" ); document.write( "This problem is a typical Math Olympiad level problem or a Math circle level problem
\n" ); document.write( "for 9-th grades high school students, who just know Algebra, but still don't know Calculus.\r
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