document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1123544>Question 1123544</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I've been trying to solve this problem for a while and would appreciate the help thank-you.\r<BR>\n" );
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document.write( "The one-to-one function g and h are defined as:\r<BR>\n" );
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document.write( "g(x)=x-9/5\r<BR>\n" );
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document.write( "h={(-7,-1),(-4,4),(-1,3),(7,1)}\r<BR>\n" );
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document.write( "Find:\r<BR>\n" );
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document.write( "g^-1(x) =\r<BR>\n" );
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document.write( "(g &#8728; g^-1)(-2) = \r<BR>\n" );
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document.write( "h^-1(-1) = <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 #739980 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=739980\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> g(x) = x - 9/5\r<BR>\n" );
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document.write( "let y = g(x)\r<BR>\n" );
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document.write( "you get y = x - 9/5\r<BR>\n" );
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document.write( "replace y with x and x with y to get:\r<BR>\n" );
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document.write( "x = y - 9/5\r<BR>\n" );
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document.write( "solve for y in terms to x to get:\r<BR>\n" );
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document.write( "y = x + 9/5\r<BR>\n" );
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document.write( "replace y with g^-1(x)\r<BR>\n" );
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document.write( "you have g(x) = x - 9/5 and g^-1(x) = x + 9/5\r<BR>\n" );
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document.write( "g^-1(x) is the inverse function of g(x).\r<BR>\n" );
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document.write( "put them on a graph and you will see that they are reflections about the x-axis.\r<BR>\n" );
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document.write( "this means that the point (x,y) in g(x) will be directly opposite the point (y,x) in g^-1(x) and equidistant from the line y = x.\r<BR>\n" );
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document.write( "here's my graph, using the point (3.4) in g(x) which is reflected about the line y = x to become the point (1.6,3.4) in g^-1(x).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/091905.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "g(x) = x - 9/5.<BR>\n" );
document.write( "when x = 3.4, g(x) becomes 3.4  - 9/5 = 1.6\r<BR>\n" );
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document.write( "g^-1(x) = x + 9/5.<BR>\n" );
document.write( "when x = 1.6, g^-1(x) becomes 1.6 + 9/5 = 3.4\r<BR>\n" );
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document.write( "the point (3.4,1.6) is a reflection of the point (1.6,3.4) about the line y = x.\r<BR>\n" );
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document.write( "this means the the point (3.4,1.6) is the same vertical distance from the line y = x as the point (1.6,3.4).\r<BR>\n" );
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document.write( "the dashed line y = -x was used to create this vertical connection through the dashed line y = x.\r<BR>\n" );
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document.write( "here's a reference on inverse functions.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.mathsisfun.com/sets/function-inverse.html\" target = \"_blank\">https://www.mathsisfun.com/sets/function-inverse.html</a>\r<BR>\n" );
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document.write( "the other property of an inverse function to note is that:\r<BR>\n" );
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document.write( "g(g^-1(x)) = x and g^-1(g(x)) = x\r<BR>\n" );
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document.write( "g(x) = x - 9/5<BR>\n" );
document.write( "g^-1(x) = x + 9/5<BR>\n" );
document.write( "g(g^-1(x)) = g(x + 9/5) = (x + 9/5) - 9/5 = x<BR>\n" );
document.write( "g^-1(g(x)) = g^-1(x - 9/5) = (x - 9/5) + 9/5 = x\r<BR>\n" );
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document.write( "the function gets you somewhere.<BR>\n" );
document.write( "the inverse function gets you back.\r<BR>\n" );
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document.write( "for example if f = (1,2), the f^-1) = (2,1)\r<BR>\n" );
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document.write( "the function f gives you y = 2 when x = 1.<BR>\n" );
document.write( "the function f^-1) gives you y = 1 when x = 2.\r<BR>\n" );
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document.write( "there are restrictions as noted in the reference.\r<BR>\n" );
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document.write( "to answer your questions:\r<BR>\n" );
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document.write( "g(x)=x-9/5 <BR>\n" );
document.write( "h={(-7,-1),(-4,4),(-1,3),(7,1)} \r<BR>\n" );
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document.write( "find g^-1(x):\r<BR>\n" );
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document.write( "we did that above.\r<BR>\n" );
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document.write( "set y = x - 9/5<BR>\n" );
document.write( "replace y with x and x with y to get x = y - 9/5<BR>\n" );
document.write( "solve for y to get y = x + 9/5<BR>\n" );
document.write( "g^-1(x) = x + 9/5\r<BR>\n" );
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document.write( "find (g &#8728; g^-1)(-2):\r<BR>\n" );
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document.write( "g(x) = x - 9/5<BR>\n" );
document.write( "g^-1(x) = x + 9/5\r<BR>\n" );
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document.write( "(g &#8728; g^-1)(x) = g(g^-1(x))<BR>\n" );
document.write( "g(x) = x - 9/5<BR>\n" );
document.write( "g^-1(x) = x + 9/5<BR>\n" );
document.write( "g(g^-1(x)) = g(x + 9/5) = (x + 9/5) - 9/5 = x<BR>\n" );
document.write( "you replace x in g(x) = x - 9/5 with g^-1(x) = x + 9/5 to get g(x + 9/5) = (x + 9/5) - 9/5 = x<BR>\n" );
document.write( "this is as it should be since, by definition:<BR>\n" );
document.write( "g(g^-1(x)) = x and g^-1(g(x)) = x\r<BR>\n" );
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document.write( "the problem says find (g &#8728; g^-1)(-2):\r<BR>\n" );
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document.write( "g^-1(-2) = (-2) + 9/5 = -2 + 9/5 = -10/5 + 9/5 = -1/5\r<BR>\n" );
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document.write( "g(g^-1(-2)) becomes g(-1/5) which becomes (-1/5) - 9/5 which becomes -10/5 which becomes -2.\r<BR>\n" );
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document.write( "therefore, (g &#8728; g^-1)(-2) = -2\r<BR>\n" );
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document.write( "here's a review on functions.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.mathsisfun.com/sets/function.html\" target = \"_blank\">https://www.mathsisfun.com/sets/function.html</a>\r<BR>\n" );
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document.write( "here's a review on functional notation.\r<BR>\n" );
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document.write( "find h^-1(-1):\r<BR>\n" );
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document.write( "you are given that:\r<BR>\n" );
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document.write( "h={(-7,-1),(-4,4),(-1,3),(7,1)} \r<BR>\n" );
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document.write( "when x = -7, y = -1<BR>\n" );
document.write( "when x = -4, y = 4<BR>\n" );
document.write( "when x = -1, y = 3<BR>\n" );
document.write( "when x = 7, y = 1\r<BR>\n" );
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document.write( "h^-1 is the inverse function of h.\r<BR>\n" );
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document.write( "therefore, in the inverse function, .....\r<BR>\n" );
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document.write( "when x = -1, y = -7<BR>\n" );
document.write( "when x = 4, y = -4<BR>\n" );
document.write( "when x = 3, y = -1<BR>\n" );
document.write( "when x = 1, y = 7\r<BR>\n" );
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document.write( "h = {(-7,-1),(-4,4),(-1,3),(7,1)} becomes:\r<BR>\n" );
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document.write( "h^-1 = {(-1,-7),(4,-4),(3,-1),(1,7)}\r<BR>\n" );
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document.write( "this agrees with the definition that, the point (x,y) in h(x) becomes the point (y,x) in h^-1(x).\r<BR>\n" );
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document.write( "that is the point that is directly opposite and equidistant from the line y = x.\r<BR>\n" );
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document.write( "your y value in the regular function becomes your x value in the inverse function and your x value in the regular function becomes your y value in the inverse function.\r<BR>\n" );
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document.write( "that's why h^-1(-1) = -7.<BR>\n" );
document.write( "h went from -1 to -7<BR>\n" );
document.write( "h^-1 goes from -7 to -1.\r<BR>\n" );
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document.write( "i think i'm right, but i could be wrong, so, if you find something confusing to you that doesn't make any sense, email me at dtheophilis@gmail.com and i'll try to explain further and / or correct any errors as best i can.\r<BR>\n" );
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document.write( "read the references.\r<BR>\n" );
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document.write( "they should help.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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