document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=127643>Question 127643</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3. Show that the composition of two linear functions is also a linear function.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #93525 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=93525\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?m_1,m_2,b_1,b_2 \in R\">\r<BR>\n" );
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document.write( "and \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=m%5B1%5Dx%2Bb%5B1%5D ALIGN=MIDDLE  ALT=\"f%28x%29=m%5B1%5Dx%2Bb%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=m%5B2%5Dx%2Bb%5B2%5D ALIGN=MIDDLE  ALT=\"g%28x%29=m%5B2%5Dx%2Bb%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f+o+g%29%28x%29=f%28g%28x%29%29=m%5B1%5D%28m%5B2%5Dx%2Bb%5B2%5D%29%2Bb%5B1%5D ALIGN=MIDDLE  ALT=\"%28f+o+g%29%28x%29=f%28g%28x%29%29=m%5B1%5D%28m%5B2%5Dx%2Bb%5B2%5D%29%2Bb%5B1%5D\"   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=m%5B1%5Dm%5B2%5Dx%2Bm%5B1%5Db%5B2%5D%2Bb%5B1%5D ALIGN=MIDDLE  ALT=\"m%5B1%5Dm%5B2%5Dx%2Bm%5B1%5Db%5B2%5D%2Bb%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?m_1,m_2,b_1,b_2 \in R\">, this means that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?m_1*m_2\in R\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?m_1*b_2\in R\">. So this shows us that if f(x) and g(x) are linear, then (f o g)(x) is a linear function.\r<BR>\n" );
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document.write( "Now let's verify this statement the other way.  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28g+o+f%29%28x%29=g%28f%28x%29%29=m%5B2%5D%28m%5B1%5Dx%2Bb%5B1%5D%29%2Bb%5B2%5D ALIGN=MIDDLE  ALT=\"%28g+o+f%29%28x%29=g%28f%28x%29%29=m%5B2%5D%28m%5B1%5Dx%2Bb%5B1%5D%29%2Bb%5B2%5D\"   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=m%5B2%5Dm%5B1%5Dx%2Bm%5B2%5Db%5B1%5D%2Bb%5B2%5D ALIGN=MIDDLE  ALT=\"m%5B2%5Dm%5B1%5Dx%2Bm%5B2%5Db%5B1%5D%2Bb%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?m_1,m_2,b_1,b_2 \in R\">, this means that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?m_2*m_1\in R\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?m_2*b_1\in R\">. So this shows us that if f(x) and g(x) are linear, then (g o f)(x) is a linear function. </SPAN>\n" );
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