document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209613>Question 1209613</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose the domain of f is (-1,3). Define the function g by<BR>\n" );
document.write( "g(x) = f(x - 4x^2)<BR>\n" );
document.write( "What is the domain of g?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849669 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849669\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to determine the domain of g:\r<BR>\n" );
document.write( "\n" );
document.write( "1. **Understand the relationship between f and g:**\r<BR>\n" );
document.write( "\n" );
document.write( "The function g(x) is defined in terms of f. The input to f is the expression x - 4x². For g(x) to be defined, this expression must fall within the domain of f, which is (-1, 3).\r<BR>\n" );
document.write( "\n" );
document.write( "2. **Set up the inequality:**\r<BR>\n" );
document.write( "\n" );
document.write( "We need to find the values of x for which:\r<BR>\n" );
document.write( "\n" );
document.write( "-1 < x - 4x² < 3\r<BR>\n" );
document.write( "\n" );
document.write( "3. **Solve the inequalities:**\r<BR>\n" );
document.write( "\n" );
document.write( "We'll solve this as two separate inequalities:\r<BR>\n" );
document.write( "\n" );
document.write( "* **-1 < x - 4x²**<BR>\n" );
document.write( "   4x² - x - 1 < 0\r<BR>\n" );
document.write( "\n" );
document.write( "   Find the roots of 4x² - x - 1 = 0 using the quadratic formula:\r<BR>\n" );
document.write( "\n" );
document.write( "   x = (1 ± √(1 + 16))/8 = (1 ± √17)/8\r<BR>\n" );
document.write( "\n" );
document.write( "   So, (1 - √17)/8 < x < (1 + √17)/8\r<BR>\n" );
document.write( "\n" );
document.write( "* **x - 4x² < 3**<BR>\n" );
document.write( "   4x² - x + 3 > 0\r<BR>\n" );
document.write( "\n" );
document.write( "   Find the roots of 4x² - x + 3 = 0. The discriminant is 1 - 4*4*3 = -47, which is negative.  This means the quadratic is always positive, so this inequality is true for all x.\r<BR>\n" );
document.write( "\n" );
document.write( "4. **Combine the solutions:**\r<BR>\n" );
document.write( "\n" );
document.write( "Since the second inequality is true for all x, we only need to consider the first inequality.\r<BR>\n" );
document.write( "\n" );
document.write( "5. **State the domain of g:**\r<BR>\n" );
document.write( "\n" );
document.write( "The domain of g is:\r<BR>\n" );
document.write( "\n" );
document.write( "((1 - √17)/8, (1 + √17)/8)<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );