document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195187>Question 1195187</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the domain of y= f(x) = cos^-1(3x + 2) and solve the equation y = 𝜋/4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #827590 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=827590\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part 1: Finding the domain.\r<BR>\n" );
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document.write( "Recall that the smallest cosine can get is -1, and the largest is 1<BR>\n" );
document.write( "I.e. the outputs are between -1 and 1, inclusive of both endpoints\r<BR>\n" );
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document.write( "This tells us the range is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+%3C=+cos%28x%29+%3C=+1 ALIGN=MIDDLE  ALT=\"-1+%3C=+cos%28x%29+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This then leads directly to the fact that the domain of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \cos^{-1}(x)\"> is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+%3C=+x+%3C=+1 ALIGN=MIDDLE  ALT=\"-1+%3C=+x+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "The domain and range swap places when going from the original to the inverse function.<BR>\n" );
document.write( "This swap is because x and y themselves swap roles. \r<BR>\n" );
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document.write( "Therefore, the input expression 3x+2 must be between -1 and 1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+%3C=+3x%2B2+%3C=+1 ALIGN=MIDDLE  ALT=\"-1+%3C=+3x%2B2+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1-2+%3C=+3x%2B2-2+%3C=+1-2 ALIGN=MIDDLE  ALT=\"-1-2+%3C=+3x%2B2-2+%3C=+1-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtracting 2 from all sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3+%3C=+3x+%3C=+-1 ALIGN=MIDDLE  ALT=\"-3+%3C=+3x+%3C=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2F3+%3C=+3x%2F3+%3C=+-1%2F3 ALIGN=MIDDLE  ALT=\"-3%2F3+%3C=+3x%2F3+%3C=+-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Dividing all sides by 3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+%3C=+x+%3C=+-1%2F3 ALIGN=MIDDLE  ALT=\"-1+%3C=+x+%3C=+-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The inequality signs stay the same the entire time. They only would flip if we divided all sides by a negative number.\r<BR>\n" );
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document.write( "Answer: The domain as a compound inequality is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+%3C=+x+%3C=+-1%2F3 ALIGN=MIDDLE  ALT=\"-1+%3C=+x+%3C=+-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Verbally we can say x must be between -1 and -1/3, inclusive of both endpoints.<BR>\n" );
document.write( "In interval notation, the domain is [-1, -1/3]. Don't forget the square brackets.\r<BR>\n" );
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document.write( "Part 2: Solve the equation for x when y = pi/4\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y = \cos^{-1}(3\text{x}+2)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{\pi}{4} = \cos^{-1}(3\text{x}+2)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \cos\left(\frac{\pi}{4}\right) = 3\text{x}+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{\sqrt{2}}{2} = 3\text{x}+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{\sqrt{2}}{2}-2 = 3\text{x}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 3\text{x} = \frac{\sqrt{2}}{2}-2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 3\text{x} = \frac{\sqrt{2}}{2}-\frac{4}{2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 3\text{x} = \frac{\sqrt{2}-4}{2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{x} = \frac{\sqrt{2}-4}{2}*\frac{1}{3}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{x} = \frac{\sqrt{2}-4}{6}\"><BR>\n" );
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