document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=201539>Question 201539</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Listed below are 5 functions each denoted g(x) and each involving a real number <BR>\n" );
document.write( "                       x<BR>\n" );
document.write( "constant c>1. If f(x)=2  , which of these 5 functions yeilds the greatest value for f(g(x)), for all x>1?\r<BR>\n" );
document.write( "\n" );
document.write( "A. g(x)= cx<BR>\n" );
document.write( "B. g(x)= c/x<BR>\n" );
document.write( "C. g(x)= x/c<BR>\n" );
document.write( "D. g(x)= x-c<BR>\n" );
document.write( "E. g(x)= log x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #151892 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=151892\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+2%5Ex ALIGN=MIDDLE  ALT=\"f%28x%29+=+2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > means this function will take whatever you give it and raise 2 to that power. So when we give it g(x) it will raise 2 to that power: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28g%28x%29%29+=+2%5E%28%28g%28x%29%29%29 ALIGN=MIDDLE  ALT=\"f%28g%28x%29%29+=+2%5E%28%28g%28x%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So now we can see what will happen to this for the various definitions of g(x).<BR>\n" );
document.write( "A. g(x)= cx<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28cx%29+=+2%5E%28cx%29 ALIGN=MIDDLE  ALT=\"f%28cx%29+=+2%5E%28cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "B. g(x)= c/x<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28c%2Fx%29+=+2%5E%28c%2Fx%29+=+2%5E%28c%2A%281%2Fx%29%29 ALIGN=MIDDLE  ALT=\"f%28c%2Fx%29+=+2%5E%28c%2Fx%29+=+2%5E%28c%2A%281%2Fx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "C. g(x)= x/c<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2Fc%29+=+2%5E%28x%2Fc%29+=+2%5E%28x%2A%281%2Fc%29%29 ALIGN=MIDDLE  ALT=\"f%28x%2Fc%29+=+2%5E%28x%2Fc%29+=+2%5E%28x%2A%281%2Fc%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "D. g(x)= x-c<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x-c%29+=+2%5E%28%28x-c%29%29 ALIGN=MIDDLE  ALT=\"f%28x-c%29+=+2%5E%28%28x-c%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "E. g(x)= log x<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28log%28%28x%29%29%29+=+2%5E%28log%28%28x%29%29%29 ALIGN=MIDDLE  ALT=\"f%28log%28%28x%29%29%29+=+2%5E%28log%28%28x%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we just need to figure out which exponent is largest<BR>\n" );
document.write( "When c>1 and x>1 ...<BR>\n" );
document.write( "A. cx > x and cx > c<BR>\n" );
document.write( "B. 1/x is between 0 and 1 so c*(1/x) < c.<BR>\n" );
document.write( "C. 1/c is between 0 and 1 so x*(1/c) < x.<BR>\n" );
document.write( "D. x-c < x<BR>\n" );
document.write( "E. log(x) < x. This is the trickiest one to explain. If you look at graphs of y=x and y=log(x) superimposed on each other you will see that for all x-values the graph of graph of y=log(x) is below the graph of y=x. So log(x) < x for all x. (Unfortunately I can't get Algebra.com software to graph log(x). If I could I'd show you these graphs here.)<br><BR>\n" );
document.write( "In summary, when c>1 and x>1 only c*x is guaranteed to be larger than both c and x. So c*x is the largest exponent which means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28cx%29 ALIGN=MIDDLE  ALT=\"2%5E%28cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the largest which means the answer is A. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );