document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183975>Question 1183975</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If f = {(1, 2), (2, 3), (3, 4), (4, 5)}, g = {(1, -2), (3, -3), (5, -5)}, and h = {(1, 0), (2, 1), (3, 2)}, find the following and state the domain.\r<BR>\n" );
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document.write( "1.  f + g\r<BR>\n" );
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document.write( "2.  f - g\r<BR>\n" );
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document.write( "3.  f * g\r<BR>\n" );
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document.write( "4.  f / h </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814513 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=814513\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll go over problem 1.\r<BR>\n" );
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document.write( "Given functions:<BR>\n" );
document.write( "f = {(1, 2), (2, 3), (3, 4), (4, 5)}<BR>\n" );
document.write( "g = {(1, -2), (3, -3), (5, -5)}<BR>\n" );
document.write( "h = {(1, 0), (2, 1), (3, 2)}\r<BR>\n" );
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document.write( "We only need to focus on functions f and g for this problem.\r<BR>\n" );
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document.write( "The f(x) function set tells us<BR>\n" );
document.write( "f(1) = 2<BR>\n" );
document.write( "f(2) = 3<BR>\n" );
document.write( "f(3) = 4<BR>\n" );
document.write( "f(4) = 5<BR>\n" );
document.write( "In general the rule is f(x) = y where (x,y) is an element of the function f<BR>\n" );
document.write( "note that f(5) is not defined\r<BR>\n" );
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document.write( "For g(x), we have<BR>\n" );
document.write( "g(1) = -2<BR>\n" );
document.write( "g(3) = -3<BR>\n" );
document.write( "g(5) = -5<BR>\n" );
document.write( "note that g(2) and g(4) are not defined.\r<BR>\n" );
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document.write( "Now onto computing f+g<BR>\n" );
document.write( "Effectively what we do is for each input (x = 1 through x = 5), we determine the y output value by adding the outputs of f(x) and g(x)\r<BR>\n" );
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document.write( "Let's compute the output when x = 1 is the input<BR>\n" );
document.write( "(f+g)(x) = f(x)+g(x)<BR>\n" );
document.write( "(f+g)(1) = f(1)+g(1)<BR>\n" );
document.write( "(f+g)(1) = 2+(-2)<BR>\n" );
document.write( "(f+g)(1) = 0<BR>\n" );
document.write( "When x = 1 is the input, y = 0 is the output for the f+g function.<BR>\n" );
document.write( "This means the ordered pair (1,0) is part of the f+g function set.\r<BR>\n" );
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document.write( "Repeat for x = 2<BR>\n" );
document.write( "(f+g)(x) = f(x)+g(x)<BR>\n" );
document.write( "(f+g)(2) = f(2)+g(2)<BR>\n" );
document.write( "(f+g)(2) = 3+undefined<BR>\n" );
document.write( "(f+g)(2) = undefined<BR>\n" );
document.write( "The g(2) is undefined as there isn't an (x,y) ordered pair with x = 2 in the g(x) function. This undefined property carries down to the final result. So there isn't any output for f+g when the input is 2.\r<BR>\n" );
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document.write( "Repeat for x = 3<BR>\n" );
document.write( "(f+g)(x) = f(x)+g(x)<BR>\n" );
document.write( "(f+g)(3) = f(3)+g(3)<BR>\n" );
document.write( "(f+g)(3) = 4+(-3)<BR>\n" );
document.write( "(f+g)(3) = 1<BR>\n" );
document.write( "This time both f(x) and g(x) are defined for this input<BR>\n" );
document.write( "The input x = 3 leads to the output y = 1 for the f+g function<BR>\n" );
document.write( "So (3,1) is part of the f+g function set\r<BR>\n" );
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document.write( "This process is continued until we reach x = 5<BR>\n" );
document.write( "You should have the following set:<BR>\n" );
document.write( "f+g = {(1,0), (3,1)}<BR>\n" );
document.write( "Earlier we found that f+g isn't defined when x = 2, so we ignore this input. Also, the x values x = 4 and x = 5 are not included because g(x) isn't defined when x = 4 and f(x) isn't defined when x = 5. Both functions need to be defined.<BR>\n" );
document.write( "In short, we ignore any inputs that lead to undefined outputs. \r<BR>\n" );
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document.write( "The domain of f+g is simply the list of x coordinates of each (x,y) point. Therefore, the domain is {1,3}<BR>\n" );
document.write( "The domain is the set of all allowed inputs of a function.\r<BR>\n" );
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document.write( "---------------------------------------------------\r<BR>\n" );
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document.write( "<font color=red>Answers for problem 1</font><BR>\n" );
document.write( "<font color=red>f+g = {(1,0), (3,1)}</font><BR>\n" );
document.write( "<font color=red>Domain = {1,3}</font>\r<BR>\n" );
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document.write( "I'll let you tackle problems 2 through 4. They follow the same basic idea. You need to go through each input (x = 1 through x = 5) and apply the arithmetic operation shown. If a function isn't defined at a particular input, then the whole thing isn't defined. Keep in mind that division by zero isn't defined either, so make sure that h(x) is nonzero as it's in the denominator.<BR>\n" );
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