document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209616>Question 1209616</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 + 1)(x - 2)).<BR>\n" );
document.write( "What is the domain of g?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849667 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=849667\">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" );
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document.write( "**1. Understand the relationship between f and g:**\r<BR>\n" );
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document.write( "The function g(x) is defined in terms of f.  The input to f is the expression (x + 1)(x - 2).  For g(x) to be defined, this expression must fall within the domain of f, which is (-1, 3).\r<BR>\n" );
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document.write( "**2. Set up the inequality:**\r<BR>\n" );
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document.write( "We need to find the values of x for which:\r<BR>\n" );
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document.write( "-1 < (x + 1)(x - 2) < 3\r<BR>\n" );
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document.write( "**3. Solve the inequalities:**\r<BR>\n" );
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document.write( "We'll solve this as two separate inequalities:\r<BR>\n" );
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document.write( "* **-1 < (x + 1)(x - 2)**<BR>\n" );
document.write( "   -1 < x² - x - 2<BR>\n" );
document.write( "   0 < x² - x - 1\r<BR>\n" );
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document.write( "   To solve this, we find the roots of the quadratic x² - x - 1 = 0 using the quadratic formula:\r<BR>\n" );
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document.write( "   x = (1 ± √(1 + 4))/2 = (1 ± √5)/2\r<BR>\n" );
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document.write( "   So, x < (1 - √5)/2 or x > (1 + √5)/2\r<BR>\n" );
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document.write( "* **(x + 1)(x - 2) < 3**<BR>\n" );
document.write( "   x² - x - 2 < 3<BR>\n" );
document.write( "   x² - x - 5 < 0\r<BR>\n" );
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document.write( "   Again, find the roots of x² - x - 5 = 0:\r<BR>\n" );
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document.write( "   x = (1 ± √(1 + 20))/2 = (1 ± √21)/2\r<BR>\n" );
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document.write( "   So, (1 - √21)/2 < x < (1 + √21)/2\r<BR>\n" );
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document.write( "**4. Combine the solutions:**\r<BR>\n" );
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document.write( "We need to find where *both* inequalities are true.  Let's approximate the roots:\r<BR>\n" );
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document.write( "* (1 - √5)/2 ≈ -0.618<BR>\n" );
document.write( "* (1 + √5)/2 ≈ 1.618<BR>\n" );
document.write( "* (1 - √21)/2 ≈ -1.791<BR>\n" );
document.write( "* (1 + √21)/2 ≈ 2.791\r<BR>\n" );
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document.write( "Combining the inequalities, we get:\r<BR>\n" );
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document.write( "(1 - √21)/2 < x < (1 - √5)/2  *or*  (1 + √5)/2 < x < (1 + √21)/2\r<BR>\n" );
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document.write( "**5. State the domain of g:**\r<BR>\n" );
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document.write( "The domain of g is the union of the two intervals:\r<BR>\n" );
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document.write( "((1 - √21)/2, (1 - √5)/2) ∪ ((1 + √5)/2, (1 + √21)/2)<BR>\n" );
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