Algebra.Com's Answer #238480 by Edwin McCravy(20059)![\"\"](\"/images/stars/stars-13.gif\")  
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define the function F on [-7,-1] by F(x)=x^2+8\r
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document.write( "Determine whether F is one to one
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document.write( "First let's draw the graph and see if it looks like it is\r\n" );
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document.write( "one-to-one, then we'll prove whether it is or not\r\n" );
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document.write( "It looks one-to-one because it does not pass beside itself\r\n" );
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document.write( "at any point.\r\n" );
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document.write( "Let's pass some horizontal lines through it, and see if\r\n" );
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document.write( "they all pass through the graph only once.\r\n" );
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document.write( "So it looks like it passes the horizontal line test, so we \r\n" );
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document.write( "believe that it is one-to-one. However \"looking and seeing\" does not \r\n" );
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document.write( "prove anything. So let's prove that it is one-to-one.\r\n" );
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document.write( "PROOF:\r\n" );
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document.write( "Suppose, for contradiction, that it is not one-to one.\r\n" );
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document.write( "Then there exists two numbers a and b, both in the domain of \r\n" );
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document.write( "F(x), which is [-7.-1], meaning that they are both negative \r\n" );
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document.write( "since the domain contains only negative numbers, such that\r\n" );
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document.write( " F(a) = F(b)\r\n" );
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document.write( "Then\r\n" );
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document.write( " a² + 8 = b² + 8\r\n" );
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document.write( "therefore\r\n" );
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document.write( " a² - b² = 0\r\n" );
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document.write( "(a - b)(a + b) = 0\r\n" );
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document.write( "a - b = 0 or a + b = 0\r\n" );
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document.write( " a = b \r\n" );
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document.write( "If a = b then that contradicts the assumption that a and b\r\n" );
document.write( "are different numbers.\r\n" );
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document.write( "The other equation a + b = 0 is not possible because a and b\r\n" );
document.write( "are both negative numbers (since the domain [-7,1] contains \r\n" );
document.write( "only negative numbers, and the sum of two negative numbers is\r\n" );
document.write( "always negative, and never 0. Therefore we have proved that\r\n" );
document.write( "the function is one-to-one.\r\n" );
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document.write( "The range is [f(-1), f(-7)] = [(-1)²+8, (-7)²+8] = [1+8,49+8] = [9,57]\r\n" );
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document.write( "Edwin \n" );
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