document.write( "Question 1151833: Let f(x) be an even function that has period 0.3 and f(2)= 5f(-4)+2
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Algebra.Com's Answer #773717 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
A periodic function repeats over and over.  Since we are not given any values\r\n" );
document.write( "for f(x), we can only hope that f(2), f(-4), and f(7) all turn out to be repeats\r\n" );
document.write( "of the same value.\r\n" );
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document.write( "Since f(x) has period 0.3, \r\n" );
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document.write( "f(x) = f(x+0.3n) for any integer n.\r\n" );
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document.write( "Therefore,\r\n" );
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document.write( "f(-4) = f(-4+0.3n)\r\n" );
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document.write( "We wonder if there is an integer n so that f(-4) = f(2),\r\n" );
document.write( "for then we could find their common value, so to find out,\r\n" );
document.write( "we set -4+0.3n = 2\r\n" );
document.write( "          0.3n = 6\r\n" );
document.write( "             n = 6/0.3     \r\n" );
document.write( "             n = 20\r\n" );
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document.write( "YES!! There is such an integer, n=20, so f(2) = f(-4+20∙0.3) = f(-4)\r\n" );
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document.write( "f(2)= 5f(-4)+2\r\n" );
document.write( "f(2)= 5f(2)+2\r\n" );
document.write( "-4f(2)=2\r\n" );
document.write( "f(2)=-0.5\r\n" );
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document.write( "We want to find f(7),  So, as before\r\n" );
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document.write( "f(7) = f(7+0.3n) for any integer n\r\n" );
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document.write( "As before, we wonder if there is an integer n so that f(7) = f(2),\r\n" );
document.write( "for then we could find their common value, so to find out,\r\n" );
document.write( "we set 7+0.3n = 2\r\n" );
document.write( "         0.3n = -5\r\n" );
document.write( "            n = -5/0.3     \r\n" );
document.write( "            n = -50/3\r\n" );
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document.write( "Oh shucks! That's not an integer, so it looks at first like we're stuck!\r\n" );
document.write( "But wait! All may not be lost!  Think now! What was given that we haven't used?\r\n" );
document.write( "Aha! Eureka! To the rescue, we were given that the function is even, and that\r\n" );
document.write( "means\r\n" );
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document.write( "f(-x) = f(x) for all x.\r\n" );
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document.write( "Therefore f(2) = f(-2)\r\n" );
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document.write( "So it will be just as good if there is an integer n so that f(7) = f(-2),\r\n" );
document.write( "for then we could find their common value, so to find out if there is such an\r\n" );
document.write( "integer, we set 7+0.3n = -2\r\n" );
document.write( "                  0.3n = -9\r\n" );
document.write( "                     n = -9/0.3     \r\n" );
document.write( "                     n = -30\r\n" );
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document.write( "Hurray! -30 is an integer. So that means \r\n" );
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document.write( "f(7) = f(7-30∙0.3) = f(-2) = f(2) = -0.5   <--the answer\r\n" );
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document.write( "Edwin
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