document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=261456>Question 261456</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help on solving this equation. I'm supposed to find the domain and range. I think that domain is x and range is y and that f(x) is the same thing as y.  Therefore, I think that f(x) is the range and x is the domain.  Here is the problem: f(x) = 2x + 4\r<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 #192679 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=192679\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "Not quite.  The domain of a function is the set of values that the independent variable (typically, and in this case, the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> such that the expression defining the function is defined.  Generally you would want to ask what is the domain of the function over the Real Numbers (or some other specific set of numbers in some cases)\r<BR>\n" );
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document.write( "In the case of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x)\ =\ 2x\ +\ 4\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> can take on any real number value and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x)\"> will take on a corresponding real number value.\r<BR>\n" );
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document.write( "Hence, the domain is the set of all real numbers.  Symbolically, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \{x\,|\,x\,\in\,R\}\">\r<BR>\n" );
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document.write( "The range is the set of values that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x)\"> takes on for every possible value in the domain.  In the case of your example, the range is also all real numbers.\r<BR>\n" );
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document.write( "Contrast this with a function such as:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ \frac{x}{x\,-\,1}\">\r<BR>\n" );
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document.write( "Here the value 1 must be excluded from the domain because if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 1\">, the denominator is zero and that makes the expression undefined.  For this example, the domain is all real numbers EXCEPT 1.\r<BR>\n" );
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document.write( "Coincidentally, the range also excludes 1 for this example.  Notice that no matter how large <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> becomes, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> will never equal <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\,-\,1\">, so the fraction can never take on the exact value of 1.\r<BR>\n" );
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document.write( "Here's another example:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ \sqrt{x\ -\ 1}\">\r<BR>\n" );
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document.write( "In this case we have to exclude any value for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> that would cause the radicand to be smaller than zero since the square root of a negative number is undefined in the real numbers.  Hence the domain is found by\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ -\ 1\ \geq\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ \geq\ 1\">\r<BR>\n" );
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document.write( "The range in this case is all positive numbers.\r<BR>\n" );
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document.write( "Of interest is the absolute value function:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ |x|\ =\ \left\{\ \ x\text{ if }x\ \geq\ 0\cr-x\text{ if }x\ <\ \,0\right\">\r<BR>\n" );
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document.write( "Which has a domain of all reals, but a range of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x)\ \geq\ 0\">\r<BR>\n" );
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document.write( "Just as a general set of rules:\r<BR>\n" );
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document.write( "Any polynomial function has a domain of all real numbers.  Polynomial functions of odd degree have ranges of all real numbers.  Polynomial functions of even degree have either a minimum or maximum value which restricts their range.\r<BR>\n" );
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document.write( "Rational functions have their domain restricted by eliminating values that make a denominator equal zero.\r<BR>\n" );
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document.write( "Functions with radicals having a even index (square root, 4th root, etc.) are restricted to values of the independent variable that make the radicand greater than or equal to zero.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sin\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \cos\"> have unrestricted domains but the range is restricted to values between -1 and 1 inclusive.\r<BR>\n" );
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document.write( "Exponential functions can take any real number as an input, but have a range strictly greater than zero.\r<BR>\n" );
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document.write( "Logarithmic functions have a domain restricted to reals strictly greater than zero.  Furthermore, the base must be strictly greater than zero.  The range is all real numbers.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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