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 #846165 by mccravyedwin(406)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"A%28x%29+=+4x%2Asqrt%281+-+x%5E2%29\"\r\n" );
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document.write( "We find the x-intercept(s) by setting A(x) = 0\r\n" );
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document.write( "\"0+=+4x%2Asqrt%281+-+x%5E2%29\"\r\n" );
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document.write( "\"4x=0\";   \"sqrt%281-x%5E2%29=0\"\r\n" );
document.write( " \"x=0\";   \"1-x%5E2=0\"\r\n" );
document.write( "               \"1=x%5E2\"\r\n" );
document.write( "               \"%22%22%2B-1=x\"\r\n" );
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document.write( "So the x-intercepts are (-1,0), (0,0), (1,0\r\n" );
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document.write( "We find the y-intercept(s) by setting x = 0  \r\n" );
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document.write( "\"A%280%29+=+4%280%29%2Asqrt%281+-+%280%29%5E2%29\"\r\n" );
document.write( "\"A%280%29+-+0\"\r\n" );
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document.write( "So the y-intercept is (0,0) which is both an x- and a y-intercept.\r\n" );
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document.write( "\"A%28x%29+=+4x%2Asqrt%281+-+x%5E2%29\"\r\n" );
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document.write( "Since a square root's radicand cannot be negative\r\n" );
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document.write( "\"1-x%5E2%3E=0\"\r\n" );
document.write( "\"%281-x%29%281%2Bx%29%3E=0\"\r\n" );
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document.write( "So either both factor are positive or both negative\r\n" );
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document.write( "\"matrix%281%2C3%2C1-x%3E=0%2CAND%2C1%2Bx%3E=0%29\" OR \"matrix%281%2C3%2C1-x%3C=0%2CAND%2C1%2Bx%3C=0%29\"\r\n" );
document.write( "\"matrix%281%2C3%2C1%3E=x%2CAND%2Cx%3E=-1%29\" OR \"cross%28matrix%281%2C3%2C1%3C=x%2CAND%2Cx%3C-1%29%29\"\r\n" );
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document.write( "So the domain is \"-1%3C=x%3C=1\" or in interval notation [-1,1]\r\n" );
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document.write( "For the range, we must find the maxima and minima\r\n" );
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document.write( "\"A%28x%29+=+4x%2Asqrt%281+-+x%5E2%29\"\r\n" );
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document.write( "\"A%28x%29+=+4%2A%28x%2A%281+-+x%5E2%29%5E%28%221%2F2%22%29%5E%22%22%29\"\r\n" );
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document.write( "We can factor the -1/2 power out of both terms and since \"1%2F2-%28-1%2F2%29=1\",\r\n" );
document.write( "we will just have the 1st power of the parenthetical expression:\r\n" );
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document.write( "Now I'll let you simplify that down to\r\n" );
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document.write( "\"dA%2Fdx=%22%22+%2B-+%284-8x%5E2%29%2Fsqrt%281-x%5E2%29\"\r\n" );
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document.write( "Set that = 0 and you get the x-coordinates of the maximum and minimum points\r\n" );
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document.write( "\"4-8x%5E2=0\"\r\n" );
document.write( "\"4=8x%5E2\"\r\n" );
document.write( "\"4%2F8=x%5E2\"\r\n" );
document.write( "\"1%2F2=x%5E2\"\r\n" );
document.write( "\"%22%22+%2B-+sqrt%281%2F2%29=x\"\r\n" );
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document.write( "Substituting:\r\n" );
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document.write( "\"A%28%22%22+%2B-+1%2Fsqrt%282%29+%29\" = \"%22%22+%2B-+4%281%2Fsqrt%282%29%29%2Asqrt%281+-+%281%2Fsqrt%282%29%29%5E2%29%29\"\r\n" );
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document.write( "\"A%28%22%22+%2B-+1%2Fsqrt%282%29+%29\" = \"%28%22%22+%2B-+4%2Fsqrt%282%29%29sqrt%281+-+1%2F2%29\"\r\n" );
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document.write( "\"A%28%22%22+%2B-+1%2Fsqrt%282%29+%29\" = \"%22%22+%2B-+%284%2Fsqrt%282%29%29%281%2Fsqrt%282%29%29\"\r\n" );
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document.write( "\"A%28%22%22+%2B-+1%2Fsqrt%282%29+%29\" = \"%22%22+%2B-+4%2F2\"\r\n" );
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document.write( "\"A%28%22%22+%2B-+1%2Fsqrt%282%29+%29\" = \"%22%22+%2B-+2\"\r\n" );
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document.write( "So the range is \"-2%3C=x%3C=2\" or in interval notation [-2,2]\r\n" );
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document.write( "Edwin
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