document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1173303>Question 1173303</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the two functions: f(x)=x^2-4 and g(x)=2x^3-3x^2-5x+6<BR>\n" );
document.write( "A. A function is defined by h(x)=f(x)/g(x). Determine the domain of this function. <BR>\n" );
document.write( "B. What are the discontinuities? (remember the discontinuities includes the removable ones and the asymptotes). <BR>\n" );
document.write( "C. State the asymptotic behaviour <BR>\n" );
document.write( "D. Find all intercepts. <BR>\n" );
document.write( "E. Sketch h(x) </SPAN>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29+=+f%28x%29%2Fg%28x%29+=+%28x%5E2-4%29%2F%282x%5E3-3x%5E2-5x%2B6%29+=+%28%28x%2B2%29%28x-2%29%29%2F%28%28x-1%29%28x-2%29%282x%2B3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The critical points on the graph are at any value of x that makes a factor of the numerator or denominator equal to 0: -2, -3/2, 1, and 2.<br><BR>\n" );
document.write( "A. Domain: any value of x except those that make the denominator 0.  So all x except 1, 2, and -3/2.<br><BR>\n" );
document.write( "B. Discontinuities: wherever there is a value of x that is excluded from the domain: at x=1, x=2, and x=-3/2.<BR>\n" );
document.write( "The discontinuity at x=2 is removable (there is a hole in the graph); the discontinuities at x=1 and x=-3/2 are asymptotes.<br><BR>\n" );
document.write( "C. Asymptotic behavior: we'll come back to that after part D.<br><BR>\n" );
document.write( "D. The y-intercept is at (0,f(0)) = (0,-4/6) = (0,-2/3).<BR>\n" );
document.write( "The only x-intercept is where the numerator is 0 and the denominator is not also 0 -- at (-2,0).<br><BR>\n" );
document.write( "Asymptotic behavior: The function value is positive for all x greater than 2 (the largest critical value).  Since the only x-intercept is at x=-2, the behavior is:<BR>\n" );
document.write( "towards +infinity as x approaches 2 from the right<BR>\n" );
document.write( "towards -infinity as x approaches 2 from the left<BR>\n" );
document.write( "towards -infinity as x approaches -3/2 from the right<BR>\n" );
document.write( "towards +infinity as x approaches -3/2 from the left<br><BR>\n" );
document.write( "Note the graph is asymptotic to 0 from above as x approaches +infinity; it is asymptotic to 0 from below as x approaches -infinity<br><BR>\n" );
document.write( "E. Graph:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-5%2C5%2C-1%2C1%2C%28x%5E2-4%29%2F%282x%5E3-3x%5E2-5x%2B6%29%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-5%2C5%2C-1%2C1%2C%28x%5E2-4%29%2F%282x%5E3-3x%5E2-5x%2B6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );