document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208034>Question 1208034</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the range of f(x) = (x^2 + 2)/(x + 4) algebraically? \r<BR>\n" );
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document.write( "It is my understanding that the domain of f inverse is the range of f. \r<BR>\n" );
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document.write( "You say?  </SPAN>\n" );
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document.write( "You are correct.<BR>\n" );
document.write( "The domain of the inverse is the range of the original function.\r<BR>\n" );
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document.write( "The domain and range swap places like this because x and y swap. After the swap the goal is to solve for y.<BR>\n" );
document.write( "f(x) = (x^2+2)/(x+4)<BR>\n" );
document.write( "y = (x^2+2)/(x+4)<BR>\n" );
document.write( "x = (y^2+2)/(y+4) .... swapping x and y<BR>\n" );
document.write( "x(y+4) = y^2+2<BR>\n" );
document.write( "xy+4x = y^2+2<BR>\n" );
document.write( "y^2-xy+2-4x = 0\r<BR>\n" );
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document.write( "To finish the process of solving for y, we use the quadratic formula.<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = -x<BR>\n" );
document.write( "c = 2-4x<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"y+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   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=y+=+%28-%28-x%29%2B-sqrt%28%28-x%29%5E2-4%2A1%2A%282-4x%29%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"y+=+%28-%28-x%29%2B-sqrt%28%28-x%29%5E2-4%2A1%2A%282-4x%29%29%29%2F%282%2A1%29\"   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=y+=+%28x%2B-sqrt%28x%5E2%2B16x-8%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"y+=+%28x%2B-sqrt%28x%5E2%2B16x-8%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The presence of \"plus/minus\" indicates the inverse isn't a function. <BR>\n" );
document.write( "Notice how the original function f(x) fails the horizontal line test.\r<BR>\n" );
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document.write( "We want the radicand x^2+16x-8 to be nonnegative so we avoid the square root of a negative number.<BR>\n" );
document.write( "Apply the quadratic formula to x^2+16x-8 = 0 to find the exact solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-8-6%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"x+=+-8-6%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-8%2B6%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"x+=+-8%2B6%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Graph out y = x^2+16x-8<BR>\n" );
document.write( "The graph is a parabola that opens upward.<BR>\n" );
document.write( "The portion between the roots is when y < 0, so the radicand x^2+16x-8 is only nonnegative when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=-8-6%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"x%3C=-8-6%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=-8%2B6%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"x%3E=-8%2B6%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Therefore the range of the function f(x), as a set of inequalities, would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=-8-6%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"y%3C=-8-6%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=-8%2B6%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"y%3E=-8%2B6%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The range in interval notation would be <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C100%2C-5%2C5%2C-5%2C5%2C%0D%0Aarc%28-3%2C-1%2C3%2C5%2C140%2C220%29%2C%0D%0Alocate%28-4%2C0%2Cmatrix%281%2C3%2C-infinity%2C%22%2C%22%2C-8-6%2Asqrt%282%29%29%29%2C%0D%0Aline%28-0.7%2C1%2C-0.7%2C-3%29%2C%0D%0Aline%28-0.7%2C1%2C-1%2C1%29%2C%0D%0Aline%28-0.7%2C-3%2C-1%2C-3%29%2C%0D%0Alocate%280%2C0%2C%22U%22%29%2C%0D%0A%0D%0Aline%280.7%2C1%2C0.7%2C-3%29%2C%0D%0Aline%280.7%2C1%2C1%2C1%29%2C%0D%0Aline%280.7%2C-3%2C1%2C-3%29%2C%0D%0Alocate%281%2C0%2Cmatrix%281%2C3%2C-8%2B6%2Asqrt%282%29%2C%22%2C%22%2Cinfinity%29%29%2C%0D%0Aarc%282.3%2C-1%2C3%2C5%2C180%2B140%2C220%2B180%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The square brackets indicate we include both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-8-6%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"y=-8-6%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-8%2B6%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"y=-8%2B6%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the range.<BR>\n" );
document.write( "Curved parenthesis are <u>always</u> with either infinity. <BR>\n" );
document.write( "This is because we cannot reach infinity.<BR>\n" );
document.write( "It's not on the number line. Rather it's a concept that represents going on forever in some direction.\r<BR>\n" );
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document.write( "If you prefer decimal approximations, then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-8-6%2Asqrt%282%29+=+-16.485 ALIGN=MIDDLE  ALT=\"y=-8-6%2Asqrt%282%29+=+-16.485\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-8%2B6%2Asqrt%282%29+=+0.485 ALIGN=MIDDLE  ALT=\"y=-8%2B6%2Asqrt%282%29+=+0.485\"   DISABLED_style= \"max-width:90%; height:auto;\" > when rounding to 3 decimal places.<BR>\n" );
document.write( "The interval notation would then become (-infinity, -16.485] U [0.485, infinity)<BR>\n" );
document.write( "The \"U\" symbol represents \"set union\". Informally it means \"or\", so the possible range of values is the interval (-infinity, -16.485] OR it's the interval [0.485, infinity)<BR>\n" );
document.write( "Replace the word \"infinity\" with the infinity symbol if necessary.<BR>\n" );
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