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" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850938 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850938\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Problem 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If f(x) is an even function, then f(-x) = f(x) for all x in the domain.<BR>\n" );
document.write( "This produces symmetry about the y axis.\r<BR>\n" );
document.write( "\n" );
document.write( "Since functions g and h are even, we know that<BR>\n" );
document.write( "g(-x) = g(x) and h(-x) = h(x) \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then,<BR>\n" );
document.write( "k(x) = f(x)+g(x)+h(x)<BR>\n" );
document.write( "k(-x) = f(-x)+g(-x)+h(-x) ....... replace each x with -x<BR>\n" );
document.write( "k(-x) = f(x)+g(x)+h(x) ........ apply definitions shown above<BR>\n" );
document.write( "k(-x) = k(x)<BR>\n" );
document.write( "Therefore k(x) is <font color=red>even</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You can use a graphing tool like <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> or <a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> to help confirm the answer.<BR>\n" );
document.write( "For instance you can form these functions<BR>\n" );
document.write( "f(x) = x^2<BR>\n" );
document.write( "g(x) = 10x^4-7<BR>\n" );
document.write( "h(x) = 8x^6+3x^2<BR>\n" );
document.write( "A polynomial is an even function when all exponents are even numbers. <BR>\n" );
document.write( "Think of the -7 as -7x^0 in the g(x) function.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Problem 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If f(x) is an odd function then f(-x) = -f(x) for all x in the domain.<BR>\n" );
document.write( "g(-x) = -g(x) is the case here as well since we're told g(x) is also odd.<BR>\n" );
document.write( "h(-x) = h(x) since h(x) is even.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "k(x) = f(x)*g(x) + h(x)<BR>\n" );
document.write( "k(-x) = f(-x)*g(-x) + h(-x)<BR>\n" );
document.write( "k(-x) = -1*f(x)*(-1)*g(x) + h(x)<BR>\n" );
document.write( "k(-x) = (-1)*(-1)*f(x)*g(x) + h(x)<BR>\n" );
document.write( "k(-x) = f(x)*g(x) + h(x)<BR>\n" );
document.write( "k(-x) = k(x)<BR>\n" );
document.write( "This proves that k(x) is <font color=red>even</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Problem 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f = even function<BR>\n" );
document.write( "g,h = odd functions\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f(-x) = f(x)<BR>\n" );
document.write( "g(-x) = -g(x)<BR>\n" );
document.write( "h(-x) = -h(x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "k(x) = f(x)*( g(x)+h(x) )<BR>\n" );
document.write( "k(-x) = f(-x)*( g(-x)+h(-x) )<BR>\n" );
document.write( "k(-x) = f(x)*( -g(x)-h(x) )<BR>\n" );
document.write( "k(-x) = f(x)*(-1)*( g(x)+h(x) )<BR>\n" );
document.write( "k(-x) = -f(x)*( g(x)+h(x) )<BR>\n" );
document.write( "k(-x) = -k(x)<BR>\n" );
document.write( "We have shown that k(x) is <font color=red>odd</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Problem 4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f,h = odd functions<BR>\n" );
document.write( "g = even\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f(-x) = -f(x) and h(-x) = -h(x)<BR>\n" );
document.write( "g(-x) = g(x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "k(x) = f(x)*g(x)*h(x)<BR>\n" );
document.write( "k(-x) = f(-x)*g(-x)*h(-x)<BR>\n" );
document.write( "k(-x) = -f(x)*g(x)*(-1)*h(x)<BR>\n" );
document.write( "k(-x) = (-1)*(-1)*f(x)*g(x)*h(x)<BR>\n" );
document.write( "k(-x) = f(x)*g(x)*h(x)<BR>\n" );
document.write( "k(-x) = k(x)<BR>\n" );
document.write( "k(x) is an <font color=red>even</font> function.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------------<BR>\n" );
document.write( "------------------------------------------------------------<BR>\n" );
document.write( "Answers:<BR>\n" );
document.write( "1. <font color=red>even</font><BR>\n" );
document.write( "2. <font color=red>even</font><BR>\n" );
document.write( "3. <font color=red>odd</font><BR>\n" );
document.write( "4. <font color=red>even</font><BR>\n" );
document.write( "In other words, answer 3 is the only \"odd\" while the others are \"even\".<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );