document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193790>Question 1193790</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the function f(x)= 3/x^2 what is the domain and range of f(x) (interval notation) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #825826 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=825826\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answers:<BR>\n" );
document.write( "<font color=red>Domain: (-infinity, 0) U (0, infinity)<BR>\n" );
document.write( "Range: (0, infinity)</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Explanation:<BR>\n" );
document.write( "Division by zero is not allowed. We cannot have zero in the denominator. Something like 5/0 is undefined.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The denominator x^2 cannot have x = 0 for that reason. Any other x value is allowed.<BR>\n" );
document.write( "The domain is the set of all real numbers x such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C%3E+0 ALIGN=MIDDLE  ALT=\"x+%3C%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "Expressed as a disjoint set of two intervals, we can say: x < 0 or x > 0<BR>\n" );
document.write( "The interval x < 0 is the same as -infinity < x < 0<BR>\n" );
document.write( "The interval x > 0 or 0 < x is the same as 0 < x < infinity\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can shorten -infinity < x < 0 to the interval notation of (-infinity, 0). Use parenthesis to exclude each endpoint. <BR>\n" );
document.write( "Similarly, 0 < x < infinity shortens to (0, infinity)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We then glue the two intervals to get the domain of (-infinity, 0) U (0, infinity)<BR>\n" );
document.write( "The U refers to the union operator in set mathematics.<BR>\n" );
document.write( "Imagine we had a number line stretching from -infinity to +infinity<BR>\n" );
document.write( "Then poke a hole at 0 to remove it from the domain. Anything else is valid. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The range is the set of all possible y outputs<BR>\n" );
document.write( "The numerator of y = 3/(x^2) is positive, and x^2 is always positive if x is a nonzero real number. <BR>\n" );
document.write( "Therefore, 3/(x^2) overall is always positive for any nonzero real number x. <BR>\n" );
document.write( "The range is y > 0 or 0 < y or 0 < y < infinity<BR>\n" );
document.write( "This shortens to the interval notation of (0, infinity)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To visually verify those answers, I recommend using your favorite graphing tool like Desmos or GeoGebra to graph out y = 3/(x^2) to see why the domain and range the way they are. <BR>\n" );
document.write( "Take note of the horizontal asymptote of y = 0 and the vertical asymptote of x = 0. They are perfectly overlapped with the x and y axis respectively.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The graph is here:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.desmos.com/calculator/qj8c4oefpo\">https://www.desmos.com/calculator/qj8c4oefpo</a><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );