document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=890433>Question 890433</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The function f is defined by f(x)=-2x^2-6x+7, x<-2. Find the inverse function and state the corresponding domain and range. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #539004 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=539004\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> see the following graph.<BR>\n" );
document.write( "see below the graph for further comments.\r<BR>\n" );
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document.write( "the domain of the original equation is x <= -2<BR>\n" );
document.write( "the range of the original equation is y <= 11\r<BR>\n" );
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document.write( "the domain of the inverse equation is x <= 11<BR>\n" );
document.write( "the range of the inverse equation is y <= -2\r<BR>\n" );
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document.write( "that's what happens with inverse equations.\r<BR>\n" );
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document.write( "the domain of the original equation becomes the range of the inverse equation.<BR>\n" );
document.write( "the range of the original equation becomes the domain of the inverse equation.\r<BR>\n" );
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document.write( "the inverse function is found by solving for x and then exchanging x with y.\r<BR>\n" );
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document.write( "the inverse function is also found by exchanging x with y and then solving for y.\r<BR>\n" );
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document.write( "either way, you get x exchanged with y.\r<BR>\n" );
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document.write( "the only difference is when you do it.\r<BR>\n" );
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document.write( "with a quadratic equation, you find the inverse equation by using the completing the square method.\r<BR>\n" );
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document.write( "i'll show you how by exchanging x with y first and then solving for y.\r<BR>\n" );
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document.write( "your original equation is f(x) = -2x^2 - 6x + 7\r<BR>\n" );
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document.write( "set y = f(x) and you get:\r<BR>\n" );
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document.write( "y = -2x^2 - 6x + 7\r<BR>\n" );
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document.write( "exchange y with x and you get:\r<BR>\n" );
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document.write( "x = -2y^2 - 6y + 7\r<BR>\n" );
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document.write( "here is where you want to complete the square.\r<BR>\n" );
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document.write( "re-arrange the terms so that all the y's are on one side of the equation and all the x's and constants are on the other side of the equation.\r<BR>\n" );
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document.write( "I added 2y^2 and 6y to both sides of the equation and I subtracted x from both sides of the equation to get:\r<BR>\n" );
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document.write( "2y^2 + 6y = 7 - x\r<BR>\n" );
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document.write( "now you need to make the coefficient of the x^2 term equal to 1, so divide both sides of the equation by 2 to get:\r<BR>\n" );
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document.write( "y^2 + 3y = (7-x)/2\r<BR>\n" );
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document.write( "now take 1/2 the coefficient of the y term and complete the square to get:\r<BR>\n" );
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document.write( "(y+3/2)^2 = (7-x)/2 + (3/2)^2\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "(y+3/2)^2 = (23-2x)/4\r<BR>\n" );
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document.write( "i'll show you how the simplification on the right side of the equation was done later.\r<BR>\n" );
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document.write( "take the square root of both sides of the equation to get:\r<BR>\n" );
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document.write( "(y+3/2) = +/- sqrt((23-2x)/4)\r<BR>\n" );
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document.write( "you can simplify the right sides of the equation by taking the 4 out from under the square root sign to get:\r<BR>\n" );
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document.write( "(y+3/2) = +/- sqrt(23-2x)/2\r<BR>\n" );
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document.write( "subtract 3/2 from both sides of the equation to get:\r<BR>\n" );
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document.write( "y = -3/2 +/- sqrt(23-2x)/2\r<BR>\n" );
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document.write( "since y needs to be less than or equal to 2, we only need the bottom half of that inverse equation, which is:\r<BR>\n" );
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document.write( "y = -3/2 - sqrt(23-2x)/2\r<BR>\n" );
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document.write( "the simplification on the right side of the equation that i mentioned above that i would show you later is done as follows:\r<BR>\n" );
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document.write( "start with:\r<BR>\n" );
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document.write( "(7-x)/2 + (3/2)^2\r<BR>\n" );
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document.write( "square (3/2) to get 9/4.\r<BR>\n" );
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document.write( "expression becomes:\r<BR>\n" );
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document.write( "(7-x)/2 + 9/4\r<BR>\n" );
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document.write( "multiply (7-x)/2 by 2/2 to get:\r<BR>\n" );
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document.write( "(14-2x)/4 + 9/4\r<BR>\n" );
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document.write( "combine the numerators under the common denominator to get:\r<BR>\n" );
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document.write( "(14 - 2x + 9) / 4\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "(23 - 2x) / 4\r<BR>\n" );
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document.write( "the simplification is complete.\r<BR>\n" );
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document.write( "some editing of the graph had to be done unless I wanted to graph manually.\r<BR>\n" );
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document.write( "this was because you have to show graph the whole equation and can't arbitrarily tell the graphing software to only take it up to a certain point, like show the graph from x = - infinity to x = -2 and don't show the rest.\r<BR>\n" );
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document.write( "i did the best i could to show you only the pieces of each graph that were applicable.\r<BR>\n" );
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document.write( "if the equations are inverses of each other, they will be reflections across the line y = x.\r<BR>\n" );
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document.write( "that line was shown along with points that were directly opposite each other.\r<BR>\n" );
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document.write( "the point (-2,11) on the original equation becomes (11,-2) on the inverse equation.\r<BR>\n" );
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document.write( "the point (-4.1,-2) on the original equation becomes (-2,-4.1) on the inverse equation.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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