document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1104434>Question 1104434</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Let f(x)=x^2 -1 and g(x)= x^2 - 2, for x has the domain of all real numbers \r<BR>\n" );
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document.write( "a. Show that (f ° g)(x)=x^4 - 4x^2 + 3. <BR>\n" );
document.write( "do we do this by putting g(x) into f(x)? How would this look like?\r<BR>\n" );
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document.write( "b. How could I sketch the graph of (f ° g)(x), for 0 is < or equal to x < or equal to 2.25?<BR>\n" );
document.write( "Could I plug this into my calculator to see the graph and just copy it from there?\r<BR>\n" );
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document.write( "c. The equation (f ° g)(x)=k has exactly two solutions, for x < or equal to x < or equal to 2.25. Find the possible values of k. <BR>\n" );
document.write( "I have no idea how to do this part. Could you help? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #719202 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=719202\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. (f ° g)(x) =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22f+%5B+g+%28+x+%29+%5D%22=f%28x%5E2-2%29=%28x%5E2-2%29%5E2-1=%28x%5E4-4x%5E2%2B4%29-1=x%5E4-4x%5E2%2B3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "b. Using a graphing calculator may be intended,<BR>\n" );
document.write( "but you could still be expected to calculate and tabulate some values.<BR>\n" );
document.write( "You may want to copy the shape, but make sure your graph goes through the zeros, maxima and minima of the function.<BR>\n" );
document.write( "A graphing calculator can tells you where the zeros, the maxima, and the minima are.<BR>\n" );
document.write( "The zeros of the function are the solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4-4x%5E2%2B3=0 ALIGN=MIDDLE  ALT=\"x%5E4-4x%5E2%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4-4x%5E2%2B3=0 ALIGN=MIDDLE  ALT=\"x%5E4-4x%5E2%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4-4x%5E2=-3 ALIGN=MIDDLE  ALT=\"x%5E4-4x%5E2=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4-4x%5E2%2B4=-3%2B4 ALIGN=MIDDLE  ALT=\"x%5E4-4x%5E2%2B4=-3%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-2%29%5E2=1 ALIGN=MIDDLE  ALT=\"%28x%5E2-2%29%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2=%22+%22+%2B-+1 ALIGN=MIDDLE  ALT=\"x%5E2-2=%22+%22+%2B-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=2+%2B-+1 ALIGN=MIDDLE  ALT=\"x%5E2=2+%2B-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%5E2=1%2C%22or%22%2Cx%5E2=3%29 ALIGN=MIDDLE  ALT=\"system%28x%5E2=1%2C%22or%22%2Cx%5E2=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "As we are working only where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=x%3C=2.25 ALIGN=MIDDLE  ALT=\"0%3C=x%3C=2.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "That means the zeros of the function happen at<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%283%29=approximately1.732 ALIGN=MIDDLE  ALT=\"x=sqrt%283%29=approximately1.732\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "If you were expected not to use a graphing calculator,<BR>\n" );
document.write( "you would have to calculate derivatives, maxima and minima.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%28x%5E4-4x%5E2%2B3%29%2Fdx=4x%5E3-8x=4x%28x%5E2-2%29=4x%28x-sqrt%282%29%29%28x%2Bsqrt%282%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That first derivative is<BR>\n" );
document.write( "positive for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3Esqrt%282%29 ALIGN=MIDDLE  ALT=\"x%3Esqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , meaning that the function increases in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%3Cx%3C2.25 ALIGN=MIDDLE  ALT=\"sqrt%282%29%3Cx%3C2.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and the derivative is<BR>\n" );
document.write( "negative for<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cx%3Csqrt%282%29 ALIGN=MIDDLE  ALT=\"0%3Cx%3Csqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  meaning that the function decreases there.<BR>\n" );
document.write( "The derivative is zero for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%282%29 ALIGN=MIDDLE  ALT=\"x=sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Those are the points where the slope of the graph is zero.<BR>\n" );
document.write( "Because the function changes from decreasing to increasing at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%282%29 ALIGN=MIDDLE  ALT=\"x=sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "that means that there is a minimum at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%282%29 ALIGN=MIDDLE  ALT=\"x=sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "At that point,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=2 ALIGN=MIDDLE  ALT=\"x%5E2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=2%5E2=4 ALIGN=MIDDLE  ALT=\"x%5E2=2%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+f+%B0+g+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+f+%B0+g+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%282%29%29=4-4%2A2%2B3=4-8%2B3=-1 ALIGN=MIDDLE  ALT=\"%28sqrt%282%29%29=4-4%2A2%2B3=4-8%2B3=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We do not care what happens for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C0 ALIGN=MIDDLE  ALT=\"x%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but (f ° g)(x) is an even function, symmetrical with respect to the y-axis, because (f ° g)(-x)=(f ° g)(x).<BR>\n" );
document.write( "That ells us that at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > there is a maximum.<BR>\n" );
document.write( "Two other points that should be plotted properly in the sketch of the graph are<BR>\n" );
document.write( "(f ° g)(0) = 3 and (f ° g)(2.25) = 8.37890625 .<BR>\n" );
document.write( "The second derivative,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2%28x%5E4-4x%5E2%2B3%29%2Fdx%5E2=d%284x%5E3-8x%29%2Fdx=12x%5E2-8 ALIGN=MIDDLE  ALT=\"d%5E2%28x%5E4-4x%5E2%2B3%29%2Fdx%5E2=d%284x%5E3-8x%29%2Fdx=12x%5E2-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "tells you where the inflection points are, and how the graph curves,<BR>\n" );
document.write( "but maybe you are not expected to know that.<BR>\n" );
document.write( "The graph with the important points in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=x%3C=2.25%29 ALIGN=MIDDLE  ALT=\"0%3C=x%3C=2.25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > circled looks like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28200%2C400%2C-2.5%2C2.5%2C-1.5%2C8.5%2C%0D%0Agraph%28200%2C400%2C-2.5%2C2.5%2C-1.5%2C8.5%2Cx%5E4-4x%5E2%2B3%29%2C%0D%0Acircle%280%2C3%2C0.1%29%2Ccircle%281%2C0%2C0.1%29%2Ccircle%281.732%2C0%2C0.1%29%2C%0D%0Acircle%281.414%2C-1%2C0.1%29%2Ccircle%282.25%2C8.379%2C0.1%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( " <BR>\n" );
document.write( "c. The solutions to (f ° g)(x)=k are the x-coordinates of the points where y=(f ° g)(x)=k.<BR>\n" );
document.write( "That is where the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E4-4x%5E2%2B3 ALIGN=MIDDLE  ALT=\"y=x%5E4-4x%5E2%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > intersects the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=k ALIGN=MIDDLE  ALT=\"y=k\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Looking at the graph,<BR>\n" );
document.write( "you can see that for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3C1 ALIGN=MIDDLE  ALT=\"k%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C400%2C-2.5%2C2.5%2C-1.5%2C8.5%2Cx%5E4-4x%5E2%2B3%2C-1.2%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C400%2C-2.5%2C2.5%2C-1.5%2C8.5%2Cx%5E4-4x%5E2%2B3%2C-1.2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28y=k%29 ALIGN=MIDDLE  ALT=\"green%28y=k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a horizontal line below the graph of the function,<BR>\n" );
document.write( "and (f ° g)(x)=k has no solution:<BR>\n" );
document.write( "If k=-1, the equation is (f ° g)(x)=1, and it has only one solution for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cx%3C=2.25 ALIGN=MIDDLE  ALT=\"0%3Cx%3C=2.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C400%2C-2.5%2C2.5%2C-1.5%2C8.5%2Cx%5E4-4x%5E2%2B3%2C-1%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C400%2C-2.5%2C2.5%2C-1.5%2C8.5%2Cx%5E4-4x%5E2%2B3%2C-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3Ck%3C=3 ALIGN=MIDDLE  ALT=\"1%3Ck%3C=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28y=k%29 ALIGN=MIDDLE  ALT=\"green%28y=k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > will cross the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E4-4x%5E2%2B3 ALIGN=MIDDLE  ALT=\"y=x%5E4-4x%5E2%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "exactly twice in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cx%3C=2.25 ALIGN=MIDDLE  ALT=\"0%3Cx%3C=2.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C400%2C-2.5%2C2.5%2C-1.5%2C8.5%2Cx%5E4-4x%5E2%2B3%2C2.3%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C400%2C-2.5%2C2.5%2C-1.5%2C8.5%2Cx%5E4-4x%5E2%2B3%2C2.3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E3 ALIGN=MIDDLE  ALT=\"y%3E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C400%2C-2.5%2C2.5%2C-1.5%2C8.5%2Cx%5E4-4x%5E2%2B3%2C4.2%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C400%2C-2.5%2C2.5%2C-1.5%2C8.5%2Cx%5E4-4x%5E2%2B3%2C4.2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "there will be only one solution in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cx%3C=2.25 ALIGN=MIDDLE  ALT=\"0%3Cx%3C=2.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the possible values of k for (f ° g)(x)=k to have 2 solutions in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cx%3C=2.25 ALIGN=MIDDLE  ALT=\"0%3Cx%3C=2.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281%3Ck%3C=3%29 ALIGN=MIDDLE  ALT=\"highlight%281%3Ck%3C=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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