document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189172>Question 1189172</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (a) Let f : (-\infty,0) \cup (0,\infty) \to \mathbb{R} be defined by<BR>\n" );
document.write( "f(x) = x - \frac{1}{x}.Show that f has no inverse function.\r<BR>\n" );
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document.write( "(b) Let g : (0,\infty) \to \mathbb{R} be defined by<BR>\n" );
document.write( "g(x) = x - \frac{1}{x}.  Show that g has an inverse function.\r<BR>\n" );
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document.write( "How can you show this without graphing.<BR>\n" );
document.write( "Thank you in advance </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #820468 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=820468\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f: (-\infty, 0) \cup (0, \infty) \to \mathbb{R}\">\r<BR>\n" );
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document.write( "Effectively, the domain is the set of nonzero numbers (aka anything but zero) and the range is the set of all real numbers  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large\mathbb{R}\">\r<BR>\n" );
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document.write( "Let y = f(x)\r<BR>\n" );
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document.write( "We have the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x+-+1%2Fx ALIGN=MIDDLE  ALT=\"y+=+x+-+1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We will swap the locations of x and y, then solve for y, to determine the inverse.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x+-+1%2Fx ALIGN=MIDDLE  ALT=\"y+=+x+-+1%2Fx\"   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=x+=+y+-+1%2Fy ALIGN=MIDDLE  ALT=\"x+=+y+-+1%2Fy\"   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=x+=+%28y%5E2%29%2Fy+-+1%2Fy ALIGN=MIDDLE  ALT=\"x+=+%28y%5E2%29%2Fy+-+1%2Fy\"   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=x+=+%28y%5E2+-+1%29%2Fy ALIGN=MIDDLE  ALT=\"x+=+%28y%5E2+-+1%29%2Fy\"   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=xy+=+y%5E2+-+1 ALIGN=MIDDLE  ALT=\"xy+=+y%5E2+-+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=0+=+y%5E2-1+-+xy ALIGN=MIDDLE  ALT=\"0+=+y%5E2-1+-+xy\"   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%5E2-xy-1+=+0 ALIGN=MIDDLE  ALT=\"y%5E2-xy-1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This equation is in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ay%5E2%2Bby%2Bc+=+0 ALIGN=MIDDLE  ALT=\"ay%5E2%2Bby%2Bc+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > with a = 1, b = -x, c = -1\r<BR>\n" );
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document.write( "Apply the quadratic formula to solve for y\r<BR>\n" );
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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%281%29%28-1%29%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"y+=+%28-%28-x%29%2B-sqrt%28%28-x%29%5E2-4%281%29%28-1%29%29%29%2F%282%281%29%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%2B4%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"y+=+%28x%2B-sqrt%28x%5E2%2B4%29%29%2F%282%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%2Bsqrt%28x%5E2%2B4%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"y+=+%28x%2Bsqrt%28x%5E2%2B4%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28x-sqrt%28x%5E2%2B4%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"y+=+%28x-sqrt%28x%5E2%2B4%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We get two different results for y, which suggests we don't have a function overall.\r<BR>\n" );
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document.write( "Consider plugging in x = 1. This leads to...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28x%2Bsqrt%28x%5E2%2B4%29%29%2F%282%29=+%281%2Bsqrt%281%5E2%2B4%29%29%2F%282%29=1.618 ALIGN=MIDDLE  ALT=\"y+=+%28x%2Bsqrt%28x%5E2%2B4%29%29%2F%282%29=+%281%2Bsqrt%281%5E2%2B4%29%29%2F%282%29=1.618\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28x-sqrt%28x%5E2%2B4%29%29%2F%282%29=+%281-sqrt%281%5E2%2B4%29%29%2F%282%29+=+-0.618 ALIGN=MIDDLE  ALT=\"y+=+%28x-sqrt%28x%5E2%2B4%29%29%2F%282%29=+%281-sqrt%281%5E2%2B4%29%29%2F%282%29+=+-0.618\"   DISABLED_style= \"max-width:90%; height:auto;\" > (both are approximate)\r<BR>\n" );
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document.write( "In short, the input x = 1 leads to multiple outputs (roughly 1.618 and -0.618). This example shows that we don't have a function. \r<BR>\n" );
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document.write( "Therefore, the inverse of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+x+-+1%2Fx ALIGN=MIDDLE  ALT=\"f%28x%29+=+x+-+1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not exist when we're working with that specified domain. We'd need to make some kind of restriction in order to find the inverse.\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Refer to the last bit of part (a) above. I mentioned we need to make a restriction on the domain for an inverse to be possible. Infinitely many such restrictions can be done to allow for f(x) to have an inverse. In this case, we're focusing on the positive real numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C+x+%3C+infinity ALIGN=MIDDLE  ALT=\"0+%3C+x+%3C+infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > aka the interval <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large(0, \infty) \"> effectively half of the original domain in part (a).\r<BR>\n" );
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document.write( "You'll follow the same exact steps as in the previous part. However, since x > 0, this means we only focus on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28x%2Bsqrt%28x%5E2%2B4%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"y+=+%28x%2Bsqrt%28x%5E2%2B4%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is indeed a function. Any x input in the domain leads to one and exactly one y output in the range. \r<BR>\n" );
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document.write( "We ignore <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28x-sqrt%28x%5E2%2B4%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"y+=+%28x-sqrt%28x%5E2%2B4%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > because every y value here is negative.\r<BR>\n" );
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document.write( "The fact that x > 0 in the domain of g(x) will lead to y > 0 in the range of the inverse of g.<BR>\n" );
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