document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209864>Question 1209864</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For each of the following functions, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain. Select all the properties that apply.<BR>\n" );
document.write( "f(x) = \frac{x}{\sqrt{x^2 + 1}} + \frac{1}{x^2} - \frac{x}{x^2 - 1} </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850912 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=850912\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze the function f(x) = x/√(x² + 1) + 1/x² - x/(x² - 1) to determine its properties.\r<BR>\n" );
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document.write( "**1. Natural Domain:**\r<BR>\n" );
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document.write( "* **√(x² + 1):** x² + 1 is always positive, so the square root is defined for all real numbers.<BR>\n" );
document.write( "* **1/x²:** x ≠ 0<BR>\n" );
document.write( "* **x/(x² - 1):** x² - 1 ≠ 0 => x ≠ ±1\r<BR>\n" );
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document.write( "Therefore, the domain is all real numbers except x = -1, 0, and 1, or (-∞, -1) U (-1, 0) U (0, 1) U (1, ∞).\r<BR>\n" );
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document.write( "**2. Even or Odd:**\r<BR>\n" );
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document.write( "* **Even:** f(-x) = f(x)<BR>\n" );
document.write( "* **Odd:** f(-x) = -f(x)\r<BR>\n" );
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document.write( "Let's test f(-x):\r<BR>\n" );
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document.write( "f(-x) = -x/√(x² + 1) + 1/x² + x/(x² - 1)\r<BR>\n" );
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document.write( "Now, let's see if f(-x) = f(x) or f(-x) = -f(x):\r<BR>\n" );
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document.write( "* f(x) = x/√(x² + 1) + 1/x² - x/(x² - 1)<BR>\n" );
document.write( "* -f(x) = -x/√(x² + 1) - 1/x² + x/(x² - 1)\r<BR>\n" );
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document.write( "Comparing:\r<BR>\n" );
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document.write( "* f(-x) ≠ f(x) (Not even)<BR>\n" );
document.write( "* f(-x) ≠ -f(x) (Not odd)\r<BR>\n" );
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document.write( "Therefore, the function is **neither even nor odd**.\r<BR>\n" );
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document.write( "**3. Increasing or Decreasing:**\r<BR>\n" );
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document.write( "* To determine if the function is increasing or decreasing, we need to analyze its derivative, f'(x).\r<BR>\n" );
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document.write( "f'(x) = d/dx [ x/√(x² + 1) + 1/x² - x/(x² - 1) ]\r<BR>\n" );
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document.write( "Let's find the derivatives of each term:\r<BR>\n" );
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document.write( "* **d/dx [ x/√(x² + 1) ]:**<BR>\n" );
document.write( "    * Using the quotient rule: [(√(x² + 1) * 1) - (x * (1/2)(x² + 1)^(-1/2)(2x))] / (x² + 1)<BR>\n" );
document.write( "    * = (√(x² + 1) - x² / √(x² + 1)) / (x² + 1)<BR>\n" );
document.write( "    * = (x² + 1 - x²) / (x² + 1)^(3/2) = 1 / (x² + 1)^(3/2)\r<BR>\n" );
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document.write( "* **d/dx [ 1/x² ]:**<BR>\n" );
document.write( "    * -2/x³\r<BR>\n" );
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document.write( "* **d/dx [ -x/(x² - 1) ]:**<BR>\n" );
document.write( "    * -[(x² - 1) * 1 - x * 2x] / (x² - 1)²<BR>\n" );
document.write( "    * -[x² - 1 - 2x²] / (x² - 1)²<BR>\n" );
document.write( "    * -[-x² - 1] / (x² - 1)² = (x² + 1) / (x² - 1)²\r<BR>\n" );
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document.write( "Therefore,\r<BR>\n" );
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document.write( "f'(x) = 1 / (x² + 1)^(3/2) - 2/x³ + (x² + 1) / (x² - 1)²\r<BR>\n" );
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document.write( "Analyzing f'(x) is complex. Due to the various terms and the domain, it's difficult to make a general statement about increasing or decreasing behavior across the entire domain. A graphing calculator or more in-depth calculus analysis would be needed.\r<BR>\n" );
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document.write( "**4. Invertibility:**\r<BR>\n" );
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document.write( "* Due to the function not being strictly increasing or decreasing across its entire domain, and the breaks in the domain, it is likely **not invertible**.\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "* **Domain:** (-∞, -1) U (-1, 0) U (0, 1) U (1, ∞)<BR>\n" );
document.write( "* **Neither even nor odd**<BR>\n" );
document.write( "* **Invertible:** No (likely)<BR>\n" );
document.write( "* **Increasing/Decreasing:** Difficult to determine without further analysis (graphing is recommended).<BR>\n" );
document.write( " </SPAN>\n" );
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