document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=193570>Question 193570</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you please help me with the following problem:\r<BR>\n" );
document.write( "\n" );
document.write( "Find the domain of f(g(x)). F(x)= 4/(x-2) and g(x)=3/(x-5)\r<BR>\n" );
document.write( "\n" );
document.write( "I know that the domain of f(x) is (-infinity,2)U (2,infinity) and g(x) is (-infinity,5)U (5,infinity) <BR>\n" );
document.write( "I know that f(x) has got to be set to the number that the domain cannot use in g(x)<BR>\n" );
document.write( "So that would be 5=4/(x-2)<BR>\n" );
document.write( "Then multiply each side by (x-2). (x-2)(5)=(4/(x-2))(x-2) this cancel out the denominator.\r<BR>\n" );
document.write( "\n" );
document.write( "Then the problem would be 5x-10=4, subtract 10 from each side to isolate the variable.  5x-10+10=4+10 => 5x=14. Then to isolate x, x would have to be divided by 5. 5x/5=14/5 => x=14/5\r<BR>\n" );
document.write( "\n" );
document.write( "So would the domain be (-infinity,2)U(2,(14/5))U((14/5),5)U(5,infinity)?\r<BR>\n" );
document.write( "\n" );
document.write( "please let me know if any mistakes were made\r<BR>\n" );
document.write( "\n" );
document.write( "thanks!<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #145291 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=145291\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, let's find the domain of g(x):\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=3%2F%28x-5%29 ALIGN=MIDDLE  ALT=\"g%28x%29=3%2F%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given function\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-5=0 ALIGN=MIDDLE  ALT=\"x-5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator equal to zero\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 5 to both sides. So plugging 5 into the denominator results in a division by zero. So we must take this value out of the domain.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the domain of g(x) is: x can be any real number but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E5 ALIGN=MIDDLE  ALT=\"x%3C%3E5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Now let's find the domain of f(g(x))\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=4%2F%28x-2%29 ALIGN=MIDDLE  ALT=\"f%28x%29=4%2F%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given function\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=4%2F%283%2F%28x-5%29-2%29 ALIGN=MIDDLE  ALT=\"f%28x%29=4%2F%283%2F%28x-5%29-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=3%2F%28x-5%29 ALIGN=MIDDLE  ALT=\"g%28x%29=3%2F%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%284%28x-5%29%29%2F%283-2%28x-5%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%284%28x-5%29%29%2F%283-2%28x-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply EVERY term by the inner LCD <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-5 ALIGN=MIDDLE  ALT=\"x-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to clear out the inner fractions.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%284x-20%29%2F%283-2x%2B10%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%284x-20%29%2F%283-2x%2B10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%284x-20%29%2F%28-2x%2B13%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%284x-20%29%2F%28-2x%2B13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%2B13=0 ALIGN=MIDDLE  ALT=\"-2x%2B13=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator equal to zero\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x=0-13 ALIGN=MIDDLE  ALT=\"-2x=0-13\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x=-13 ALIGN=MIDDLE  ALT=\"-2x=-13\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms on the right side.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-13%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"x=%28-13%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=13%2F2 ALIGN=MIDDLE  ALT=\"x=13%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So we must take <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=13%2F2 ALIGN=MIDDLE  ALT=\"x=13%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the domain (since this value causes a division by zero)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the domain of f(g(x)) is: x can be any real number but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E13%2F2 ALIGN=MIDDLE  ALT=\"x%3C%3E13%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In addition, we made a previous statement that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E5 ALIGN=MIDDLE  ALT=\"x%3C%3E5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So this means that the domain of f(g(x)) in interval notation is \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(-\infty,5\right)\cup\left(5,\frac{13}{2}\right)\cup\left(\frac{13}{2},\infty\right)\"> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );