document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209855>Question 1209855</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) = sqrt{10 - x} + \sqrt{x + 10} </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850889 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=850889\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze the function f(x) = √(10 - x) + √(x + 10).\r<BR>\n" );
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document.write( "**1. Natural Domain:**\r<BR>\n" );
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document.write( "* **√(10 - x):** 10 - x ≥ 0 => x ≤ 10<BR>\n" );
document.write( "* **√(x + 10):** x + 10 ≥ 0 => x ≥ -10\r<BR>\n" );
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document.write( "Therefore, the domain is the intersection of these two conditions: -10 ≤ x ≤ 10, or [-10, 10].\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) = √(10 - (-x)) + √((-x) + 10)<BR>\n" );
document.write( "f(-x) = √(10 + x) + √(10 - x)\r<BR>\n" );
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document.write( "Comparing:\r<BR>\n" );
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document.write( "* f(-x) = √(10 + x) + √(10 - x)<BR>\n" );
document.write( "* f(x) = √(10 - x) + √(10 + x)\r<BR>\n" );
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document.write( "Therefore, f(-x) = f(x), and the function is **even**.\r<BR>\n" );
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document.write( "Since the function is even, it cannot be 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) = (10 - x)^(1/2) + (x + 10)^(1/2)\r<BR>\n" );
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document.write( "Now, find the derivative:\r<BR>\n" );
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document.write( "f'(x) = (1/2)(10 - x)^(-1/2)(-1) + (1/2)(x + 10)^(-1/2)(1)<BR>\n" );
document.write( "f'(x) = -1 / (2√(10 - x)) + 1 / (2√(x + 10))\r<BR>\n" );
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document.write( "To find when f'(x) > 0 (increasing) and f'(x) < 0 (decreasing), we need to set f'(x) = 0:\r<BR>\n" );
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document.write( "-1 / (2√(10 - x)) + 1 / (2√(x + 10)) = 0<BR>\n" );
document.write( "1 / (2√(x + 10)) = 1 / (2√(10 - x))<BR>\n" );
document.write( "√(x + 10) = √(10 - x)<BR>\n" );
document.write( "x + 10 = 10 - x<BR>\n" );
document.write( "2x = 0<BR>\n" );
document.write( "x = 0\r<BR>\n" );
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document.write( "Now, let's test intervals:\r<BR>\n" );
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document.write( "* **For -10 ≤ x < 0:**<BR>\n" );
document.write( "    * Let's test x = -9:<BR>\n" );
document.write( "    * f'(-9) = -1 / (2√(10 - (-9))) + 1 / (2√(-9 + 10))<BR>\n" );
document.write( "    * f'(-9) = -1 / (2√19) + 1 / (2√1)<BR>\n" );
document.write( "    * f'(-9) ≈ -0.114 + 0.5<BR>\n" );
document.write( "    * f'(-9) > 0 (increasing)\r<BR>\n" );
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document.write( "* **For 0 < x ≤ 10:**<BR>\n" );
document.write( "    * Let's test x = 9:<BR>\n" );
document.write( "    * f'(9) = -1 / (2√(10 - 9)) + 1 / (2√(9 + 10))<BR>\n" );
document.write( "    * f'(9) = -1 / (2√1) + 1 / (2√19)<BR>\n" );
document.write( "    * f'(9) ≈ -0.5 + 0.114<BR>\n" );
document.write( "    * f'(9) < 0 (decreasing)\r<BR>\n" );
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document.write( "Therefore, the function is increasing on [-10, 0] and decreasing on [0, 10].\r<BR>\n" );
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document.write( "**4. Invertibility:**\r<BR>\n" );
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document.write( "* Since the function is not strictly increasing or decreasing over its entire domain, it is **not invertible**.\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "* **Domain:** [-10, 10]<BR>\n" );
document.write( "* **Even**<BR>\n" );
document.write( "* **Invertible:** No<BR>\n" );
document.write( "* **Increasing:** [-10, 0]<BR>\n" );
document.write( "* **Decreasing:** [0, 10]<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );