document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168075>Question 1168075</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let f(x)=√42−x and g(x)=x²−x. \r<BR>\n" );
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document.write( "Give the domain of f ∘ g using interval notation. \r<BR>\n" );
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document.write( "I can't figure this one yet, please!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #792739 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=792739\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x) = sqrt(42 - x)\r<BR>\n" );
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document.write( "g(x) = x^2 - x\r<BR>\n" );
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document.write( "fog(x) = f(g(x)) = f(x^2-x) = (42 - (x^2 - x))\r<BR>\n" );
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document.write( "you are replacing the x in f(x) with x^2 - x from g(x).\r<BR>\n" );
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document.write( "to get a real answer, the expression under the square root sign has to be greater than or equal to 0.\r<BR>\n" );
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document.write( "in order for that to happen, 42 - (x^2 - x) must be greater than or equal to 0.\r<BR>\n" );
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document.write( "your inequality becomes 42 - (x^2 - x) >= 0\r<BR>\n" );
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document.write( "simplify to get 42 - x^2 + x >= 0\r<BR>\n" );
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document.write( "order the terms in descending order of degree to get = -x^2 + x + 42 >= 0\r<BR>\n" );
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document.write( "solve this quadratic equation for -x^2 + x + 42 = 0\r<BR>\n" );
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document.write( "to factor, multiply both sides of the equation by -1 to get x^2 - x + 42 = 0\r<BR>\n" );
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document.write( "factor to get (x - 7) * (x + 6) = 0\r<BR>\n" );
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document.write( "solve for x to get x = 7 or x = -6\r<BR>\n" );
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document.write( "when x = 7, 42 - (x^2 - x) becomes 42 - (49 - 7) which becomes 42 - 42 which becomes 0.\r<BR>\n" );
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document.write( "when x = -6, 42 - (x^2 - x) becomes 42 - (36 + 6) which becomes 42 - 42 which becomes 0.\r<BR>\n" );
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document.write( "you get 42 - (x^2 - x) = 0 when x = 7 or when x = -6\r<BR>\n" );
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document.write( "you want it to be greater than 0 as well.\r<BR>\n" );
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document.write( "when x = 6 (less than 7), 42 - (x^2 - x) becomes 42 - (36 - 6) which becomes 42 - 30 which is greater than 0.\r<BR>\n" );
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document.write( "this means x <= 7 is good.\r<BR>\n" );
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document.write( "when x = -5 (greater than -6), 42 - (x^2 - x) becomes 42 - (25 + 5) which becomes 42 - 30 which is greater than 0.\r<BR>\n" );
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document.write( "this means x >= -6 is good.\r<BR>\n" );
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document.write( "it appears that you will get a real solution when -6 <= x <= 7.\r<BR>\n" );
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document.write( "that should be your domain.\r<BR>\n" );
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document.write( "your equation can be graphed as shown below.\r<BR>\n" );
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document.write( "the domain is -6 <= x <= 7\r<BR>\n" );
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document.write( "the range is 0 <= y <= 6.5\r<BR>\n" );
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document.write( "that's because sqrt(42 - sqrt(x^2 -x)) will be at a maximum when (x^2 - x) is at a minimum.\r<BR>\n" );
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document.write( "x^2 - x is a quadratic equation whose minimum will be when x = -b/2a<BR>\n" );
document.write( "a is the coefficient of the x^2 term and b is the coefficient of the x term.<BR>\n" );
document.write( "a = 1 and b = -1, so x = -b/2a becomes x = 1/2 = .5\r<BR>\n" );
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document.write( "x^2 - x will be at a minimum when x = .5\r<BR>\n" );
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document.write( "when x = .5, sqrt(42 - (x^2 - x)) becomes sqrt(42 - (.5^2 - .5)) which becomes sqrt(42 - (.25 - .5)) which becomes sqrt(42 - (-.25)) which becomes sqrt(42 + .25) which becomes sqrt(42.25) which is equal to 6.5.\r<BR>\n" );
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document.write( "the maximum y value is 6.5\r<BR>\n" );
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document.write( "the range is 0 <= y <= 6.5\r<BR>\n" );
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document.write( "you have:\r<BR>\n" );
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document.write( "the domain is -6 <= x <= 7 and the range is 0 <= y <= 6.5\r<BR>\n" );
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document.write( "in interval notation, the domain is x = [-6,7].\r<BR>\n" );
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document.write( "if fog(x) is a good equation, then you should get the same answer as when you solve for g(x) and then use the answer from g(x) as input to f(x)\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "when x = 5, g(x) = x^2 - x becomes g(x) = 25 - 5 which becomes g(x) = 20\r<BR>\n" );
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document.write( "g(x) becomes x in f(x).\r<BR>\n" );
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document.write( "when x = 20, f(x) = sqrt(42 - x) becomes f(x) = sqrt(42 - 20) which becomes f(x) = sqrt(22)\r<BR>\n" );
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document.write( "when x = 5, fog(x) = sqrt(42 - (x^2 - x)) becomes sqrt(42 - (25 - 5)) which becomes sqrt(42 - 20) which becomes sqrt(22).\r<BR>\n" );
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document.write( "you get the same value whether you use the fog(x) equation or whether you first find g(x) and then use the result from g(x) as input to f(x).\r<BR>\n" );
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document.write( "this confirms the solution is correct.\r<BR>\n" );
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