document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1087149>Question 1087149</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the function f(x)= |x-1|+|x-2|\r<BR>\n" );
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document.write( "Show that f(x)= -2x+3, if x &#8804; 1<BR>\n" );
document.write( "1 if 1 < x < 2<BR>\n" );
document.write( "2x-3 if x &#8805; 2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #701422 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=701422\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">Given function: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+abs%28x-1%29%2Babs%28x-2%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+abs%28x-1%29%2Babs%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+abs%28x-1%29 ALIGN=MIDDLE  ALT=\"g%28x%29+=+abs%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29+=+abs%28x-2%29 ALIGN=MIDDLE  ALT=\"h%28x%29+=+abs%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+g%28x%29%2Bh%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+g%28x%29%2Bh%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We can write the absolute value functions, g(x) and h(x), into piecewise function form<BR>\n" );
document.write( "For g(x) we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/ZLRN3mr.png\"><BR>\n" );
document.write( "which basically says <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+-%28x-1%29 ALIGN=MIDDLE  ALT=\"g%28x%29+=+-%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+1 ALIGN=MIDDLE  ALT=\"x+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > OR <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+x-1 ALIGN=MIDDLE  ALT=\"g%28x%29+=+x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E+1 ALIGN=MIDDLE  ALT=\"x+%3E+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "and for h(x) we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/u3qFiX9.png\"><BR>\n" );
document.write( "which basically says <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29+=+-%28x-2%29 ALIGN=MIDDLE  ALT=\"h%28x%29+=+-%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+2 ALIGN=MIDDLE  ALT=\"x+%3C=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > OR <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29+=+x-2 ALIGN=MIDDLE  ALT=\"h%28x%29+=+x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > if <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( "We have three intervals <BR>\n" );
document.write( "Interval A: the interval where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+1 ALIGN=MIDDLE  ALT=\"x+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Interval B: the interval where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1+%3C+x+%3C+2 ALIGN=MIDDLE  ALT=\"+1+%3C+x+%3C+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Interval C: the interval where <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( "Focus on interval A. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+1 ALIGN=MIDDLE  ALT=\"x+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > here this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+-%28x-1%29 ALIGN=MIDDLE  ALT=\"g%28x%29+=+-%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=-%28x-2%29 ALIGN=MIDDLE  ALT=\"h%28x%29=-%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Therefore, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+g%28x%29%2Bh%28x%29=-%28x-1%29%2B%28-%28x-2%29%29=-x%2B1-x%2B2=-2x%2B3 ALIGN=MIDDLE  ALT=\"f%28x%29+=+g%28x%29%2Bh%28x%29=-%28x-1%29%2B%28-%28x-2%29%29=-x%2B1-x%2B2=-2x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+%3C=+1 ALIGN=MIDDLE  ALT=\"+x+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In short, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-2x%2B3 ALIGN=MIDDLE  ALT=\"f%28x%29=-2x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+%3C=+1 ALIGN=MIDDLE  ALT=\"+x+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now move onto interval B. We have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1+%3C+x+%3C+2 ALIGN=MIDDLE  ALT=\"+1+%3C+x+%3C+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This interval will have us use g(x) = x-1 and h(x) = -(x-2). Add the functions to get <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+g%28x%29%2Bh%28x%29=x-1%2B%28-%28x-2%29%29=x-1-x%2B2=1 ALIGN=MIDDLE  ALT=\"f%28x%29+=+g%28x%29%2Bh%28x%29=x-1%2B%28-%28x-2%29%29=x-1-x%2B2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1+%3C+x+%3C+2 ALIGN=MIDDLE  ALT=\"+1+%3C+x+%3C+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In short, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=1 ALIGN=MIDDLE  ALT=\"f%28x%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1+%3C+x+%3C+2 ALIGN=MIDDLE  ALT=\"+1+%3C+x+%3C+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Finally interval C. Now <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;\" >. For this interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=x-1 ALIGN=MIDDLE  ALT=\"g%28x%29=x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=x-2 ALIGN=MIDDLE  ALT=\"h%28x%29=x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+g%28x%29%2Bh%28x%29=x-1%2Bx-2=2x-3 ALIGN=MIDDLE  ALT=\"f%28x%29+=+g%28x%29%2Bh%28x%29=x-1%2Bx-2=2x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > 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;\" ><BR>\n" );
document.write( "In short, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x-3 ALIGN=MIDDLE  ALT=\"f%28x%29=2x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > 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;\" >\r<BR>\n" );
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document.write( "To recap, we found the following<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-2x%2B3 ALIGN=MIDDLE  ALT=\"f%28x%29=-2x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+%3C=+1 ALIGN=MIDDLE  ALT=\"+x+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=1 ALIGN=MIDDLE  ALT=\"f%28x%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1+%3C+x+%3C+2 ALIGN=MIDDLE  ALT=\"+1+%3C+x+%3C+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x-3 ALIGN=MIDDLE  ALT=\"f%28x%29=2x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > 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;\" ><BR>\n" );
document.write( "Which can be combined to form the piecewise function<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/GJnHPjY.png\"><BR>\n" );
document.write( "So this proves that overall, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+k%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+k%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all real numbers x\r<BR>\n" );
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document.write( "As visual confirmation, <a  rel=nofollow HREF=\"https://i.imgur.com/zw0U3Fj.png\">here</a> is a graph showing a side by side comparision of f(x) and k(x). I let f(x) = |x-1|+|x-2| while k(x) is the piecewise function equivalent form. The two function curves are blue and brown respectively. As you can see, they are identical. If you laid them on the same xy coordinate system then you'd have the same graph. One function curve would overlap the other perfectly.  <BR>\n" );
document.write( "Note: The graph was created with <a  rel=nofollow HREF= \"https://www.geogebra.org/home?ggbLang=en\">GeoGebra</a> (free graphing software).</font> </SPAN>\n" );
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