document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202361>Question 1202361</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The function g(x) = 4 - 3sinx is defined for the domain π/2 ≤ x ≤ A<BR>\n" );
document.write( "(i)State the largest value of A for which g has an inverse<BR>\n" );
document.write( "(ii) For this value of A, find the value of g^-1(3) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #837132 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3835)</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=837132\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (i)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Make sure your calculator is in radian mode.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The period of sin(x) is 2pi units. The curve repeats itself every 2pi units.<BR>\n" );
document.write( "Half of this is 2pi/2 = pi, and this amount is added onto the left endpoint pi/2. <BR>\n" );
document.write( "This is so we can determine the largest value of A to have g(x) be invertible.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A = pi/2 + pi = pi/2 + 2pi/2 = (pi+2pi)/2 = 3pi/2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore, g(x) is invertible on the interval pi/2 ≤ x ≤ 3pi/2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Check out this Desmos graph<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/7uxfzjmjsk\">https://www.desmos.com/calculator/7uxfzjmjsk</a><BR>\n" );
document.write( "The value of A is currently set to 3pi/2 (which is 4.712389 approximately)<BR>\n" );
document.write( "If you move the slider of A around, then the blue curve will grow or shrink depending on what happens with A.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If A > 3pi/2, then the blue curve will not be one-to-one. Pieces of it will fail the horizontal line test. <BR>\n" );
document.write( "Therefore, it won't be invertible for A > 3pi/2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: <font color=red>A = 3pi/2</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=====================================================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Part (ii)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The task of finding g^(-1)(3) is the same as solving g(x) = 3 when pi/2 ≤ x ≤ 3pi/2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use your calculator to find these two approximations:<BR>\n" );
document.write( "pi/2 = 1.570796<BR>\n" );
document.write( "3pi/2 = 4.712389\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This means pi/2 ≤ x ≤ 3pi/2 approximates to 1.570796 ≤ x ≤ 4.712389\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We're looking for a value of x between roughly 1.570796 and 4.712389, that will make g(x) = 3 true.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's plug in g(x) = 3 and solve for x.<BR>\n" );
document.write( "g(x) = 4 - 3sin(x)<BR>\n" );
document.write( "3 = 4 - 3sin(x)<BR>\n" );
document.write( "-3sin(x) = 3-4<BR>\n" );
document.write( "-3sin(x) = -1<BR>\n" );
document.write( "sin(x) = -1/(-3)<BR>\n" );
document.write( "sin(x) = 1/3<BR>\n" );
document.write( "x = arcsin(1/3) or x = pi - arcsin(1/3)<BR>\n" );
document.write( "x = 0.339837 or x = 2.801756<BR>\n" );
document.write( "Make sure your calculator is in radian mode.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The first solution x = 0.339837 is not in the interval 1.570796 ≤ x ≤ 4.712389<BR>\n" );
document.write( "The second solution x = 2.801756 is in that interval, and it is the approximate final answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's introduce the horizontal line y = 3 to the graph<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/uknjchzkwd\">https://www.desmos.com/calculator/uknjchzkwd</a><BR>\n" );
document.write( "The sine curve and the straight line intersect at roughly (2.802,3) to help confirm our answer. <BR>\n" );
document.write( "It means g(2.802) = 3 approximately.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If x = 2.802, then,<BR>\n" );
document.write( "g(x) = 4 - 3sin(x)<BR>\n" );
document.write( "g(2.802) = 4 - 3sin(2.802)<BR>\n" );
document.write( "g(2.802) = 3.00069088973563<BR>\n" );
document.write( "which is fairly close to 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "and if x = 2.801756, then,<BR>\n" );
document.write( "g(x) = 4 - 3sin(x)<BR>\n" );
document.write( "g(2.801756) = 4 - 3sin(2.801756)<BR>\n" );
document.write( "g(2.801756) = 3.00000072369363<BR>\n" );
document.write( "Both results are really close to 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: <font color=red>Approximately 2.801756</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );