document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209870>Question 1209870</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Fill in the blanks.\r<BR>\n" );
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document.write( "If f(x) is an even function and g(x) is an even function and h(x) is an even function, then f(x) + g(x) + h(x) is an ___ function.\r<BR>\n" );
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document.write( "If f(x) is an odd function and g(x) is an odd function and h(x) is an even function, then f(x)*g(x) + h(x) is an ___ function.\r<BR>\n" );
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document.write( "If f(x) is an even function and g(x) is an odd function and h(x) is an odd function, then f(x)*(g(x) + h(x)) is an ___ function.\r<BR>\n" );
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document.write( "If f(x) is an odd function and g(x) is an even function and h(x) is an odd function, then f(x)*g(x)*h(x) is an ___ function.<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 #850922 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=850922\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down each scenario:\r<BR>\n" );
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document.write( "**Key Properties:**\r<BR>\n" );
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document.write( "* **Even Function:** f(-x) = f(x)<BR>\n" );
document.write( "* **Odd Function:** f(-x) = -f(x)\r<BR>\n" );
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document.write( "**1. If f(x), g(x), and h(x) are all even functions, then f(x) + g(x) + h(x) is an ___ function.**\r<BR>\n" );
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document.write( "* Let's test:<BR>\n" );
document.write( "    * [f(-x) + g(-x) + h(-x)] = f(x) + g(x) + h(x)<BR>\n" );
document.write( "* Therefore, f(x) + g(x) + h(x) is an **even** function.\r<BR>\n" );
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document.write( "**2. If f(x) and g(x) are odd functions and h(x) is an even function, then f(x)*g(x) + h(x) is an ___ function.**\r<BR>\n" );
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document.write( "* Let's test:<BR>\n" );
document.write( "    * f(-x) * g(-x) + h(-x) = (-f(x)) * (-g(x)) + h(x) = f(x) * g(x) + h(x)<BR>\n" );
document.write( "* Therefore, f(x)*g(x) + h(x) is an **even** function.\r<BR>\n" );
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document.write( "**3. If f(x) is an even function and g(x) and h(x) are odd functions, then f(x)*(g(x) + h(x)) is an ___ function.**\r<BR>\n" );
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document.write( "* Let's test:<BR>\n" );
document.write( "    * f(-x) * (g(-x) + h(-x)) = f(x) * (-g(x) - h(x)) = -f(x) * (g(x) + h(x))<BR>\n" );
document.write( "* Therefore, f(x) * (g(x) + h(x)) is an **odd** function.\r<BR>\n" );
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document.write( "**4. If f(x) and h(x) are odd functions and g(x) is an even function, then f(x)*g(x)*h(x) is an ___ function.**\r<BR>\n" );
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document.write( "* Let's test:<BR>\n" );
document.write( "    * f(-x) * g(-x) * h(-x) = (-f(x)) * g(x) * (-h(x)) = f(x) * g(x) * h(x)<BR>\n" );
document.write( "    * Now, lets test by negating all x's.<BR>\n" );
document.write( "    * f(-x) * g(-x) * h(-x) = (-f(x)) * g(x) * (-h(x)) = f(x) * g(x) * h(x)<BR>\n" );
document.write( "    * f(-x)g(-x)h(-x) = (-f(x)) * g(x) * (-h(x)) = f(x)g(x)h(x)<BR>\n" );
document.write( "    * f(-x)g(-x)h(-x) = (-1)f(x) * g(x) * (-1)h(x) = f(x)g(x)h(x)<BR>\n" );
document.write( "    * Now lets test by negating one x at a time.<BR>\n" );
document.write( "    * f(-x)g(x)h(x) = (-f(x))g(x)h(x) = -f(x)g(x)h(x)<BR>\n" );
document.write( "    * f(x)g(-x)h(x) = f(x)g(x)h(x)<BR>\n" );
document.write( "    * f(x)g(x)h(-x) = f(x)g(x)(-h(x)) = -f(x)g(x)h(x)<BR>\n" );
document.write( "    * f(-x)g(-x)h(x) = (-f(x))g(x)h(x) = -f(x)g(x)h(x)<BR>\n" );
document.write( "    * f(-x)g(x)h(-x) = (-f(x))g(x)(-h(x)) = f(x)g(x)h(x)<BR>\n" );
document.write( "    * f(x)g(-x)h(-x) = f(x)g(x)(-h(x)) = -f(x)g(x)h(x)<BR>\n" );
document.write( "    * f(-x)g(-x)h(-x) = (-f(x))g(x)(-h(x)) = f(x)g(x)h(x)<BR>\n" );
document.write( "    * The product of two odd functions is an even function, and the product of an even and an odd function is an odd function.<BR>\n" );
document.write( "    * so odd * even * odd = even * odd = odd.<BR>\n" );
document.write( "* Therefore, f(x) * g(x) * h(x) is an **odd** function.<BR>\n" );
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