document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1088255>Question 1088255</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let<BR>\n" );
document.write( "\[f(x) = \frac{3x - 7}{x + 1}.\]Find the range of $f$. Give your answer as an interval. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #702501 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=702501\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">The function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%283x-7%29%2F%28x%2B1%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%283x-7%29%2F%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has the horizontal asymptote <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%29%2Fx+=+3%2F1+=+3 ALIGN=MIDDLE  ALT=\"%283x%29%2Fx+=+3%2F1+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "You divide the leading terms where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C%3E+0 ALIGN=MIDDLE  ALT=\"x+%3C%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "As x gets larger and larger, f(x) will approach f(x) = 3.\r<BR>\n" );
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document.write( "What this means is that any value but y = 3 is possible for the range. \r<BR>\n" );
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document.write( "If we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%283x-7%29%2F%28x%2B1%29 ALIGN=MIDDLE  ALT=\"y+=+%283x-7%29%2F%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we swapped x and y, then we'd have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%283y-7%29%2F%28y%2B1%29 ALIGN=MIDDLE  ALT=\"x+=+%283y-7%29%2F%28y%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's solve for y\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%283y-7%29%2F%28y%2B1%29 ALIGN=MIDDLE  ALT=\"x+=+%283y-7%29%2F%28y%2B1%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=x%28y%2B1%29+=+3y-7 ALIGN=MIDDLE  ALT=\"x%28y%2B1%29+=+3y-7\"   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%2Bx+=+3y-7 ALIGN=MIDDLE  ALT=\"xy%2Bx+=+3y-7\"   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-3y+=+-7-x ALIGN=MIDDLE  ALT=\"xy-3y+=+-7-x\"   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-3%29+=+-7-x ALIGN=MIDDLE  ALT=\"y%28x-3%29+=+-7-x\"   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+=+%28-7-x%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"y+=+%28-7-x%29%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which is the inverse function. The denominator of this inverse, x-3, indicates that 3 is not in the domain of the inverse. This confirms that 3 is not in the range of the original function.\r<BR>\n" );
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document.write( "So the range is the set of all y values such that y is a real number but y cannot equal 3. \r<BR>\n" );
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document.write( "In set builder notation, we'd say <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left\{y|y\in \mathbb{R}, \ y \neq 3\right\}\">\r<BR>\n" );
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document.write( "In interval notation, we'd write <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(-\infty, 3\right) \cup \left(3, \infty\right)\">\r<BR>\n" );
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document.write( "The big U means \"union\" to essentially \"glue\" the two intervals together.\r<BR>\n" );
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document.write( "Here's what the graph looks like. The blue dashed line is the horizontal asymptote y = 3 (it goes through the two points (0,3) and (1,3))<BR>\n" );
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document.write( "The green curve approaches the dashed line but it will never cross it or touch it. So this shows us that any output but y = 3 is possible. \r<BR>\n" );
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