document.write( "Question 1196484: If $\"f%282a-b%29=f%28a%29f%28b%29\"$ for all a and b, and the function is never equal to zero, find the value of f(5). \n" ); document.write( "

Algebra.Com's Answer #829324 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "If f(2a-b) = f(a)*f(b) for all a and b, and the function is never equal to zero,
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\n" ); document.write( "\n" ); document.write( " Let solve the problem in 4 (four) steps.\r
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document.write( "(a)  Let a = 1, b = 1.  Then, according to the given formula, \r\n" );
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document.write( "     f(2*1-1) = f(1)*f(1),    or\r\n" );
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document.write( "     f(1)     = f(1)*f(1). \r\n" );
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document.write( "     Since f(1) =/= 0  (as it is given),  it implies  f(1) = 1   (we can cancel the common factor f(1) in both sides).\r\n" );
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document.write( "(b)  Let a = 1, b = 2.  Then, according to the given formula, \r\n" );
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document.write( "     f(2*1-1) = f(1)*f(2),    or\r\n" );
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document.write( "     f(1)     = f(1)*f(2).\r\n" );
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document.write( "     Since f(1) = 1 (we just know it from (a)),  it implies f(2) = 1.\r\n" );
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document.write( "(c)  Let a = 2, b = 1.  Then, according to the given formula, \r\n" );
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document.write( "     f(2*2-1) = f(2)*f(1),    or\r\n" );
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document.write( "     f(3)     = 1*1 = 1      (since we just know from (b) and (a) that f(2) = 1, f(1) = 1 ).\r\n" );
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document.write( "(d)  Let a = 3, b = 1.  Then, according to the given formula, \r\n" );
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document.write( "     f(2*3-1) = f(3)*f(1),    or\r\n" );
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document.write( "     f(5)     = 1*1 = 1      (since we just know from (c) and (a) that f(3) = 1, f(1) = 1 ).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Thus we just found out that f(5) = 1, and the solution is complete at this point.\r
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