document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199856>Question 1199856</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the inverse function of f(x) = √(x+2)+2 </SPAN>\n" );
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document.write( "<font color=red>Answer:</font> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f^{-1}(\text{x}) = (\text{x}-2)^2-2\"><BR>\n" );
document.write( "Domain of inverse:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+2 ALIGN=MIDDLE  ALT=\"x+%3E=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Range of inverse:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+-2 ALIGN=MIDDLE  ALT=\"y+%3E=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The graph is shown below.\r<BR>\n" );
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document.write( "Work Shown:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(\text{x}) = \sqrt{\text{x}+2}+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{y} = \sqrt{\text{x}+2}+2\"> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{x} = \sqrt{\text{y}+2}+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{x}-2 = \sqrt{\text{y}+2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (\text{x}-2)^2 = \text{y}+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (\text{x}-2)^2-2 = \text{y}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{y} = (\text{x}-2)^2-2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f^{-1}(\text{x}) = (\text{x}-2)^2-2\">\r<BR>\n" );
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document.write( "Explanation:<BR>\n" );
document.write( "Replace f(x) with y. Swap x and y as shown in step 3. Then solve for y to determine the inverse.\r<BR>\n" );
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document.write( "The domain of f(x) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+-2 ALIGN=MIDDLE  ALT=\"x+%3E=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is to avoid a negative value under the square root.<BR>\n" );
document.write( "This means the range of the inverse is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+-2 ALIGN=MIDDLE  ALT=\"y+%3E=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The domain and range swap roles when going from original function to inverse, and vice versa, simply because of the swap of x and y mentioned earlier.\r<BR>\n" );
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document.write( "The range of f(x) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+2 ALIGN=MIDDLE  ALT=\"y+%3E=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This means the domain of the inverse is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+2 ALIGN=MIDDLE  ALT=\"x+%3E=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-5%2C5%2C-5%2C5%2C-100%2Csqrt%28x%2B2%29%2B2%2C%28sqrt%28x-2%29%2Fsqrt%28x-2%29%29%2A%28%28x-2%29%5E2-2%29%2C-100%2C-100%2C-100%2C%28sqrt%28sin%285%2Ax%29%29%2Fsqrt%28sin%285%2Ax%29%29%29%2Ax%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The original f(x) function is in green.<BR>\n" );
document.write( "The inverse function is in blue.<BR>\n" );
document.write( "To go from the green curve to the blue curve, and vice versa, reflect over the dashed line y = x.\r<BR>\n" );
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document.write( "Note that the inverse is a half-parabola since it is only drawn when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+2 ALIGN=MIDDLE  ALT=\"x+%3E=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (domain of the inverse).\r<BR>\n" );
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document.write( "Graphing tools such as Desmos and GeoGebra are very useful as quick visual verification methods.<BR>\n" );
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