document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=218207>Question 218207</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can someone help me with this please. IF f(x)=3x+1, and g(x)=(x^*2+5x)^*-1/2, find g(f(x)). Any explanation on how to do it would be appreciated.. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #164366 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=164366\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This would look like the following if my assumption about what you are writing are correct:<BR>\n" );
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document.write( "Whether my assumptions are correct or not, the concept is the same and you can apply it to the correct formula if I got it wrong.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the function is a set of rules that operate on the argument to get a result.<BR>\n" );
document.write( "If you say f(x) = x^2, then the function name is f and the argument is x and the set of rules are to square the argument to get a result.<BR>\n" );
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document.write( "If you let x = 3, then f(x) = x^2 becomes f(3) = (3)^2<BR>\n" );
document.write( "the name of the function is the same,<BR>\n" );
document.write( "the argument changes from x to 3.<BR>\n" );
document.write( "the set of rules is the same and operates on the argument which in this case is 3 rather than x.<BR>\n" );
document.write( "the only thing you changed is the argument.  everything else remains the same.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "now change the argument to (x^3)<BR>\n" );
document.write( "f(x) = x^2 becomes f(x^3) = (x^3)^2 = x^6<BR>\n" );
document.write( "the name of the function is the same.<BR>\n" );
document.write( "the set of rules is the same.<BR>\n" );
document.write( "the argument has changed so the same set of rules is being applied to the new argument of (x^3) rather than x.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "this is exactly what you are doing when you take g(f(x))<BR>\n" );
document.write( "the argument that was x is being replaced with f(x)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "your equations are:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "f(x)=3x+1<BR>\n" );
document.write( "and:<BR>\n" );
document.write( "g(x)=(x^2+5x)^(-1/2)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "your first function is f(x) = 3x+1.<BR>\n" );
document.write( "the argument is x and the function being performed is to take 3 times it and add 1 to it.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "your second function is g(x) = (x^2 + 5x)^(-1/2)<BR>\n" );
document.write( "the argument is x and the function being performed is to square it and then add 5 times it and then raise the sum of them to the (-1/2) power.<BR>\n" );
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document.write( "when you take g(f(x)), then the argument becomes f(x) rather than x.  Since f(x) = (3x+1), then the new argument is (3x+1) and the function being performed remains the same except it is operating on (3x+1) rather than x.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the function g(x) operating on the argument x is:<BR>\n" );
document.write( "g(x) = (x^2 + 5x)^(-1/2)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the function g(x) operating on the argument (3x+1) is:<BR>\n" );
document.write( "g(f(x)) = g(3x+1) = ((3x+1)^2 + 5*(3x+1))^(-1/2)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "These formulas look like the following after putting them through the formula generating functions of algebra.com:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=3x%2B1 ALIGN=MIDDLE  ALT=\"f%28x%29=3x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=%28x%5E2%2B5x%29%5E%28-1%2F2%29 ALIGN=MIDDLE  ALT=\"g%28x%29=%28x%5E2%2B5x%29%5E%28-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28f%28x%29%29+=+g%283x%2B1%29+=+%28%283x%2B1%29%5E2+%2B+5%2A%283x%2B1%29%29%5E%28-1%2F2%29 ALIGN=MIDDLE  ALT=\"g%28f%28x%29%29+=+g%283x%2B1%29+=+%28%283x%2B1%29%5E2+%2B+5%2A%283x%2B1%29%29%5E%28-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "This is your answer and should be correct, but you can simplify it further if you wish.<BR>\n" );
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document.write( "This means to remove parentheses and combine like terms and take the negative out of the exponent I believe.<BR>\n" );
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document.write( "We'll do that and see what we get.<BR>\n" );
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document.write( "your equation of:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28f%28x%29%29+=+g%283x%2B1%29+=+%28%283x%2B1%29%5E2+%2B+5%2A%283x%2B1%29%29%5E%28-1%2F2%29 ALIGN=MIDDLE  ALT=\"g%28f%28x%29%29+=+g%283x%2B1%29+=+%28%283x%2B1%29%5E2+%2B+5%2A%283x%2B1%29%29%5E%28-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%283x%2B1%29+=+1%2F%289x%5E2+%2B+21x+%2B+6%29%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"g%283x%2B1%29+=+1%2F%289x%5E2+%2B+21x+%2B+6%29%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "How do we know we did this right?<BR>\n" );
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document.write( "take any value of x and place it in the original equations and then place it in the final equation and see if you get the same answer.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "let's take x = 7<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+3x%2B1 ALIGN=MIDDLE  ALT=\"f%28x%29+=+3x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "when x = 7, then:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%287%29+=+3%2A7+%2B+1+=+22 ALIGN=MIDDLE  ALT=\"f%287%29+=+3%2A7+%2B+1+=+22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+%28x%5E2+%2B+5x%29%5E%28-1%2F2%29 ALIGN=MIDDLE  ALT=\"g%28x%29+=+%28x%5E2+%2B+5x%29%5E%28-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "when x = 22, then:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%2822%29+=+%28%2822%29%5E2+%2B+5%2A22%29%5E%28-1%2F2%29+=+%28484+%2B+110%29%5E%28-1%2F2%29+=+.041030497 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28f%28x%29%29+=+%28%283x%2B1%29%5E2+%2B+5%2A%283x%2B1%29%29%5E%28-1%2F2%29 ALIGN=MIDDLE  ALT=\"g%28f%28x%29%29+=+%28%283x%2B1%29%5E2+%2B+5%2A%283x%2B1%29%29%5E%28-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "when x = 7, then:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28f%287%29%29+=+%28%283%2A7%2B1%29%5E2+%2B+5%2A%283%2A7%2B1%29%29%5E%28-1%2F2%29 ALIGN=MIDDLE  ALT=\"g%28f%287%29%29+=+%28%283%2A7%2B1%29%5E2+%2B+5%2A%283%2A7%2B1%29%29%5E%28-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28f%287%29%29+=+%28%2822%29%5E2+%2B+5%2A%2822%29%29%5E%28-1%2F2%29 ALIGN=MIDDLE  ALT=\"g%28f%287%29%29+=+%28%2822%29%5E2+%2B+5%2A%2822%29%29%5E%28-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28f%287%29%29+=+%28%2822%29%5E2+%2B+5%2A22%29%5E%28-1%2F2%29+=+%28484+%2B+110%29%5E%28-1%2F2%29+=+.041030497 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "The answers come out the same so the formulas are equivalent.<BR>\n" );
document.write( "g(x) when x = 22 is the same as g(f(x)) when x = 7.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "Your answer is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%283x%2B1%29+=+1%2F%289x%5E2+%2B+21x+%2B+6%29%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"g%283x%2B1%29+=+1%2F%289x%5E2+%2B+21x+%2B+6%29%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > simplified as far as I can take it.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "That same answer is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28f%28x%29%29+=+g%283x%2B1%29+=+%28%283x%2B1%29%5E2+%2B+5%2A%283x%2B1%29%29%5E%28-1%2F2%29 ALIGN=MIDDLE  ALT=\"g%28f%28x%29%29+=+g%283x%2B1%29+=+%28%283x%2B1%29%5E2+%2B+5%2A%283x%2B1%29%29%5E%28-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > before the simplification process was done.<BR>\n" );
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document.write( "</TD></TR></TABLE>\n" );