document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=828013>Question 828013</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need finding the composition of functions (f o g)(x).?\r<BR>\n" );
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document.write( "1. g(x) = 3/(x - 1), f(x) = (x - 1)/(x - 3) \r<BR>\n" );
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document.write( "A. (x - 1)/(x - 3) * 3/(x - 1) <BR>\n" );
document.write( "B. (x - 1)/(x - 1) * 3/(x - 1) <BR>\n" );
document.write( "C. (x - 4)/(x - 2) <BR>\n" );
document.write( "D. (x - 4)/(3x - 3) <BR>\n" );
document.write( "E. (x - 4)/(3x - 6) <BR>\n" );
document.write( "F. 3x^2 \r<BR>\n" );
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document.write( "2. g(x) = x2 - 8, f(x) = (-x + 1)^1/2 \r<BR>\n" );
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document.write( "A. (-x2 + 9)^(1/2) <BR>\n" );
document.write( "B. (-x2 - 8)^(1/2) <BR>\n" );
document.write( "C. (-x + 1)^(1/2) (x2 - 8) <BR>\n" );
document.write( "D. (-x + 1)^(1/2) (x2 - 8) <BR>\n" );
document.write( "E. (-x2 - 9)^(1/2) <BR>\n" );
document.write( "F. (x - 2)^2 \r<BR>\n" );
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document.write( "3. g(x) = (x2 -16)^(1/2), f(x) = (3 - x)^(1/2) \r<BR>\n" );
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document.write( "A. (3 - x)^(1/2), (x2 -16)^(1/2) <BR>\n" );
document.write( "B. (-x-13)^(1/2) <BR>\n" );
document.write( "C. (3-(x2 -16)^(1/4)) <BR>\n" );
document.write( "D. (3x2 + 16) <BR>\n" );
document.write( "E. (3x(1/2) + 16) <BR>\n" );
document.write( "F. (3-(x2 -16)^(1/2))^(1/2) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #498974 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=498974\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First let's review function notation. In<BR>\n" );
document.write( "f(x)<ul><li>The expression between the parentheses represents the input to the function. Here the input is an x. f(4) on the other hand indicates that the input is a 4.</li><li>The whole expression, \"f(x)\" represents the output of the function (when the input is x).</li></ul><BR>\n" );
document.write( "(f o g)(x) is another way of writing f(g(x)). In the parentheses for the function f we see \"g(x)\". This means that we will be using g(x) as input to function f:<BR>\n" );
document.write( "(f o g)(x)<BR>\n" );
document.write( "f(g(x))<BR>\n" );
document.write( "Substituting g(x) in we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28+3%2F%28x+-+1%29%29 ALIGN=MIDDLE  ALT=\"f%28+3%2F%28x+-+1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3%2F%28x+-+1%29 ALIGN=MIDDLE  ALT=\"+3%2F%28x+-+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as input to f we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28%283%2F%28x+-+1%29%29+-+1%29%2F%28%283%2F%28x+-+1%29%29+-+3%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28%283%2F%28x+-+1%29%29+-+1%29%2F%28%283%2F%28x+-+1%29%29+-+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is not one of the provided answers. So we need to start simplifying until we get a match. To eliminate the \"little\" fractions within the \"big\" fraction we multiply the numerator and denominator of the \"big\" fraction by the lowest common denominator (LCD) of the \"little\" denominators. Since both \"little\" denominators are (x-1), the LCD is (x-1):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28%28%283%2F%28x+-+1%29%29+-+1%29%2F%28%283%2F%28x+-+1%29%29+-+3%29%29%28%28x-1%29%2F%28x-1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To multiply we will need to use the Distributive Property:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28%283%2F%28x+-+1%29%29%28x-1%29+-+1%28x-1%29%29%2F%28%283%2F%28x+-+1%29%29%28x-1%29+-+3%28x-1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28%283%2Fcross%28%28x+-+1%29%29%29cross%28%28x-1%29%29+-+x%2B1%29%2F%28%283%2Fcross%28%28x+-+1%29%29%29cross%28%28x-1%29%29+-+3x%2B3%29 ALIGN=MIDDLE   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+=+%283+-+x%2B1%29%2F%283+-+3x%2B3%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%283+-+x%2B1%29%2F%283+-+3x%2B3%29\"   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+=+%284+-+x%29%2F%286+-+3x%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%284+-+x%29%2F%286+-+3x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is still not one of the given answers. But it is close to answer E. In fact the our numerator, 4-x, and E's numerator, x-4, are opposites of each other. And so are the two denominators! So if we multiply the numerator and denominator by -1:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28%284+-+x%29%2F%286+-+3x%29%29%28%28-1%29%2F%28-1%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28%284+-+x%29%2F%286+-+3x%29%29%28%28-1%29%2F%28-1%29%29\"   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+=+%28-4+%2B+x%29%2F%28-6+%2B+3x%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28-4+%2B+x%29%2F%28-6+%2B+3x%29\"   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+=+%28x+%2B+%28-4%29%29%2F%283x%2B%28-6%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28x+%2B+%28-4%29%29%2F%283x%2B%28-6%29%29\"   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+=+%28x+-4%29%2F%283x-6%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28x+-4%29%2F%283x-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and we have answer E!<br><BR>\n" );
document.write( "I'll leave the other two for you to finish. (Please do not include so many problems, especially ones that take substantial effort, in a single post!) </SPAN>\n" );
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