document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186928>Question 1186928</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For each of the following statements, explain why it is true for all functions f and g or an example of specific functions f and g showing that it is false.\r<BR>\n" );
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document.write( "a. All elements in the domain of g must be in the domain of f o g (fog)<BR>\n" );
document.write( "b. All elements in the range of f must be in the range of f o g </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849326 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=849326\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze each statement:\r<BR>\n" );
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document.write( "**a. All elements in the domain of *g* must be in the domain of *f o g***\r<BR>\n" );
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document.write( "**True.**\r<BR>\n" );
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document.write( "*Explanation:* The composition *f o g* (or *f(g(x))*) is defined as applying the function *g* first, and then applying the function *f* to the result.  For *f(g(x))* to be defined, *g(x)* *must* be defined.  This means that *x* must be in the domain of *g*.  If *x* is not in the domain of *g*, then *g(x)* is undefined, and therefore *f(g(x))* is also undefined.  So, any *x* in the domain of *f o g* *must* also be in the domain of *g*.\r<BR>\n" );
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document.write( "**b. All elements in the range of *f* must be in the range of *f o g***\r<BR>\n" );
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document.write( "**False.**\r<BR>\n" );
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document.write( "*Counterexample:*\r<BR>\n" );
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document.write( "Let *f(x) = x²*.  The range of *f* is [0, ∞).\r<BR>\n" );
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document.write( "Let *g(x) = x + 1*.\r<BR>\n" );
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document.write( "Then *(f o g)(x) = f(g(x)) = f(x + 1) = (x + 1)²*. The range of *f o g* is also [0, ∞).\r<BR>\n" );
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document.write( "However, let's consider a slightly different g(x)\r<BR>\n" );
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document.write( "Let *f(x) = x²*.  The range of *f* is [0, ∞).\r<BR>\n" );
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document.write( "Let *g(x) = 0*.\r<BR>\n" );
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document.write( "Then *(f o g)(x) = f(g(x)) = f(0) = 0²=0*. The range of *f o g* is just {0}.\r<BR>\n" );
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document.write( "In this case, the range of *f* is [0, ∞), but the range of *f o g* is only {0}.  Therefore, not all elements in the range of *f* are in the range of *f o g*.\r<BR>\n" );
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document.write( "In general, the range of *f o g* is a *subset* of the range of *f*. It can be smaller, but not larger.<BR>\n" );
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