document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194670>Question 1194670</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. For the functions f(x) = 2^x and g(x) = 3^x :\r<BR>\n" );
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document.write( "a) What are the Domains and Ranges of these functions?\r<BR>\n" );
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document.write( "b) What point(s) do they have in common and why?\r<BR>\n" );
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document.write( "c) Starting with y = 2^x, what would y = 2^x+4  -  3 look like in comparison to y = 2^x? Explain each step and show each interim equation which illustrates each step. You may attach a graph if you'd like.\r<BR>\n" );
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document.write( "d) How does y = (1/2)^x look in relation to y = 2^x? Why is this? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #826918 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=826918\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Both functions are monotonically increasing -- an increase in x means an increase in y.<br><BR>\n" );
document.write( "a) A positive number raised to any real number power is valid expression; and it is always positive.  So the domain of both functions is all real numbers, and the range of both functions is y>0.<br><BR>\n" );
document.write( "b) Since both functions are monotonically increasing, they will intersect at only one point.  Since any number to the 0 power is equal to 1, the single point of intersection is (0,1).<br><BR>\n" );
document.write( "c) I will assume by \"2^x+4-3\" what you mean is \"2^(x+4)-3\" = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28x%2B4%29-3 ALIGN=MIDDLE  ALT=\"2%5E%28x%2B4%29-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The \"x+4\" shifts the graph 4 units to the left; then the \"-3\" shifts it 3 units down.<br><BR>\n" );
document.write( "Here is a graph:<BR>\n" );
document.write( "y=2^x (red),<BR>\n" );
document.write( "y=2^(x+4) (green) (red, shifted 4 to the left),<BR>\n" );
document.write( "y=2^(x+4)-3 (blue) (green, shifted 3 down)<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28800%2C400%2C-6%2C6%2C-6%2C16%2C2%5Ex%2C2%5E%28x%2B4%29%2C2%5E%28x%2B4%29-3%29 ALIGN=MIDDLE  ALT=\"graph%28800%2C400%2C-6%2C6%2C-6%2C16%2C2%5Ex%2C2%5E%28x%2B4%29%2C2%5E%28x%2B4%29-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "d) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%5Ex+=+%282%5E%28-1%29%29%5Ex+=+2%5E%28-x%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29%5Ex+=+%282%5E%28-1%29%29%5Ex+=+2%5E%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That means the graph of y=(1/2)x is the reflection in the y axis of y=2^x.<br><BR>\n" );
document.write( "Here is a graph:<BR>\n" );
document.write( "y=2^x (red);<BR>\n" );
document.write( "y=(1/2)^x=2^(-x) (green)<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-4%2C4%2C-2%2C18%2C2%5Ex%2C%281%2F2%29%5Ex%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-4%2C4%2C-2%2C18%2C2%5Ex%2C%281%2F2%29%5Ex%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );