document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209854>Question 1209854</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) = x^3 - 3x^2 + 10x - 15  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850890 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=850890\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze the function f(x) = x³ - 3x² + 10x - 15.\r<BR>\n" );
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document.write( "**1. Natural Domain:**\r<BR>\n" );
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document.write( "* Since f(x) is a polynomial, its natural domain is all real numbers, (-∞, ∞).\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)³ - 3(-x)² + 10(-x) - 15<BR>\n" );
document.write( "f(-x) = -x³ - 3x² - 10x - 15\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) = x³ - 3x² + 10x - 15<BR>\n" );
document.write( "* -f(x) = -x³ + 3x² - 10x + 15\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) = 3x² - 6x + 10\r<BR>\n" );
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document.write( "To find when f'(x) > 0 (increasing) and when f'(x) < 0 (decreasing), we need to analyze the discriminant of the quadratic f'(x):\r<BR>\n" );
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document.write( "Discriminant (Δ) = b² - 4ac<BR>\n" );
document.write( "Δ = (-6)² - 4(3)(10)<BR>\n" );
document.write( "Δ = 36 - 120<BR>\n" );
document.write( "Δ = -84\r<BR>\n" );
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document.write( "Since the discriminant is negative, the quadratic f'(x) has no real roots. Also, since the coefficient of x² in f'(x) is positive (3), the parabola opens upwards. This means f'(x) is always positive for all real numbers x.\r<BR>\n" );
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document.write( "Therefore, f'(x) > 0 for all x, and the function f(x) is **always increasing** on its domain.\r<BR>\n" );
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document.write( "**4. Invertibility:**\r<BR>\n" );
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document.write( "* If a function is strictly increasing or strictly decreasing, it is invertible.<BR>\n" );
document.write( "* Since the function is strictly increasing over its domain, it is **invertible**.\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "* **Domain:** (-∞, ∞)<BR>\n" );
document.write( "* **Neither even nor odd**<BR>\n" );
document.write( "* **Invertible:** Yes<BR>\n" );
document.write( "* **Increasing:** Yes<BR>\n" );
document.write( "* **Decreasing:** No<BR>\n" );
document.write( " </SPAN>\n" );
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