document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194424>Question 1194424</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 4. Is f(x) - g(x) = g(x) - f(x) true for all functions? Justify your answer.<BR>\n" );
document.write( "**Writing one example will not justify your answer for ALL functions.<BR>\n" );
document.write( "Which operation property can we use to prove this is true? **<BR>\n" );
document.write( "(2 marks) 5. Is (f + g)(x) = (g + f )(x) true for all functions? Justify your answer.<BR>\n" );
document.write( "**Writing one example will not justify your answer for ALL functions. <BR>\n" );
document.write( "Which operation property can we use to prove this is true? **<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #826622 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=826622\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Problem 4\r<BR>\n" );
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document.write( "One example could be<BR>\n" );
document.write( "f(x) = x+5<BR>\n" );
document.write( "g(x) = 2x-3\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "f(x) - g(x) = (x+5) - (2x-3)<BR>\n" );
document.write( "f(x) - g(x) = x+5 - 2x+3<BR>\n" );
document.write( "f(x) - g(x) = (x-2x) + (5+3)<BR>\n" );
document.write( "f(x) - g(x) = -x + 8\r<BR>\n" );
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document.write( "On the other hand,<BR>\n" );
document.write( "g(x) - f(x) = (2x-3) - (x+5)<BR>\n" );
document.write( "g(x) - f(x) = 2x-3 - x-5<BR>\n" );
document.write( "g(x) - f(x) = (2x-x) + (-3-5)<BR>\n" );
document.write( "g(x) - f(x) = x - 8\r<BR>\n" );
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document.write( "This shows that f(x) - g(x) and g(x) - f(x) are not the same. <BR>\n" );
document.write( "There are some special cases when f(x) - g(x) = g(x) - f(x) is true for all x; however, the equation is false in general for any functions f,g.\r<BR>\n" );
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document.write( "I don't know why your teacher wrote \"Writing one example will not justify your answer for ALL functions.\" when that's exactly all we need. One example is enough to disprove the claim made. The claim is the equation works for ALL functions. But we clearly found two functions where this doesn't work. It seems like your teacher made a typo. \r<BR>\n" );
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document.write( "Problem 5\r<BR>\n" );
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document.write( "The reason why f(x)+g(x) is the same as g(x)+f(x) is the same reasoning why x+y = y+x is true.\r<BR>\n" );
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document.write( "We can add any two numbers in any order we want<BR>\n" );
document.write( "2+3 = 3+2<BR>\n" );
document.write( "7+9 = 9+7<BR>\n" );
document.write( "etc\r<BR>\n" );
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document.write( "Functions are just an extension of this idea.\r<BR>\n" );
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document.write( "Here's one example using functions<BR>\n" );
document.write( "f(x) = 2x+7<BR>\n" );
document.write( "g(x) = 3x-1<BR>\n" );
document.write( "f(x)+g(x) = (2x+7)+(3x-1) = 5x+6<BR>\n" );
document.write( "g(x)+f(x) = (3x-1)+(2x+7) = 5x+6\r<BR>\n" );
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document.write( "The 2x and 3x combine to 5x no matter what order you add them in. The 7 and -1 combine to 6.\r<BR>\n" );
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document.write( "As your instructions say, this example does not prove every single case. But it does help illustrate why it works. \r<BR>\n" );
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document.write( "The property used here is the commutative property of addition, which was the x+y = y+x mentioned.<BR>\n" );
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