document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=208474>Question 208474</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am trying to figure out a question that has 2 parts.  The question is as follows:\r<BR>\n" );
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document.write( "A)  What is the domain and range of the relation x +[y] = 3?<BR>\n" );
document.write( "b)  What is the domain and range of the function y = 2x^2-3?\r<BR>\n" );
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document.write( "Since [y] cannot be a negative number would the answer for part (a) be:  (-&, 3)<BR>\n" );
document.write( "For part (b) I am not quite sure bit I think it would be a union containing the square root of 2).\r<BR>\n" );
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document.write( "I am lost!\r<BR>\n" );
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document.write( "Any help would be greatly appreciated.\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 #157703 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=157703\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> use the || rather than the [].<BR>\n" );
document.write( "|x| means absolute value of x.<BR>\n" );
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document.write( "the | is the shift \<BR>\n" );
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document.write( "PROBLEM NUMBER 1<BR>\n" );
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document.write( "domain and range of the function x + |y| = 3<BR>\n" );
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document.write( "A)  What is the domain and range of the relation x +[y] = 3?<BR>\n" );
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document.write( "my previous answer i think was incorrect.<BR>\n" );
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document.write( "i believe the right answer is that the domain has to be x <= 3.<BR>\n" );
document.write( "the range, however, will be any real value of y simply because the absolute value of y is positive regardless if y is negative or positive.<BR>\n" );
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document.write( "here's why.<BR>\n" );
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document.write( "x + |y| = 3<BR>\n" );
document.write( "subtract x from both sides to get:<BR>\n" );
document.write( "|y| = 3 - x<BR>\n" );
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document.write( "|y| is positive so 3 - x has to be positive.<BR>\n" );
document.write( "3 - x is positive if x <= 3<BR>\n" );
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document.write( "this makes the original equation of x + |y| valid as long as x <= 3.<BR>\n" );
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document.write( "let x = 3, then 3 + |y| = 3 will be satisfied if y = 0.<BR>\n" );
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document.write( "let x = 5 which is > 3.<BR>\n" );
document.write( "this make 5 + |y| = 3 invalid because there is no value of y that will make the |y| = - 2 which it would have to be if the equation were to be satisfied  because the absolute value of y has top be positive.<BR>\n" );
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document.write( "so, my original answer is incorrect.<BR>\n" );
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document.write( "the correct ansWer is:<BR>\n" );
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document.write( "domain is all real values of x <= 3<BR>\n" );
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document.write( "with a domain of x <= 3, the range will be:<BR>\n" );
document.write( "|y| = x-3<BR>\n" );
document.write( "meaning that:<BR>\n" );
document.write( "y = 3-x<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "y = - (3-x)<BR>\n" );
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document.write( "if x = 3, then y will be 0 or -0<BR>\n" );
document.write( "if x = 0, then y will be -3 or 3<BR>\n" );
document.write( "if x is less than 0, then y will be a value > 3 or < -3<BR>\n" );
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document.write( "looks like the domain is x <= 3, and the range is all real values of y<BR>\n" );
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document.write( "let's see how that works:<BR>\n" );
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document.write( "as long as we keep the value of x less than or equal to 3 we should be ok.<BR>\n" );
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document.write( "FIRST<BR>\n" );
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document.write( "let x = 3<BR>\n" );
document.write( "this means that:<BR>\n" );
document.write( "y = 3-3 = 0<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "y = - (3-3) = -0<BR>\n" );
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document.write( "if y = 0, then 3 + |0| = 3 + 0 = 3 which satisfies the original equation of x + |y| = 3<BR>\n" );
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document.write( "if y = -0, then 3 + |-0| = 3 + 0 = 3 which satisfies the original equation of x + |y| = 3<BR>\n" );
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document.write( "SECOND<BR>\n" );
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document.write( "let x = 0<BR>\n" );
document.write( "this means that:<BR>\n" );
document.write( "y = 3-0 = 3<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "y = - (3-0) = -3<BR>\n" );
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document.write( "if y = 3, then 0 + |3|  = 0 + 3 = 3 which satisfies the original equation of x + |y| = 3<BR>\n" );
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document.write( "if y = -3, then 0 + |-3| = 0 + 3 = 3 which satisfies the original equation of x + |y| = 3<BR>\n" );
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document.write( "THIRD<BR>\n" );
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document.write( "let x = -100<BR>\n" );
document.write( "this means that:<BR>\n" );
document.write( "y = 3-(-100) = 103<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "y = - (3-(-100)) = -103<BR>\n" );
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document.write( "if y = 103, then -100 + |103|  = -100 + 103 = 3 which satisfies the original equation of x + |y| = 3<BR>\n" );
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document.write( "if y = -103, then -100 + |-103| = -100 + 103 = 3 which satisfies the original equation of x + |y| = 3<BR>\n" );
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document.write( "so far as long as we allowed x to be smaller than or equal to 3 we were in good shape.<BR>\n" );
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document.write( "if we allow x to be > 3 we should run into a problem.<BR>\n" );
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document.write( "let x = 5 which is greater than 3<BR>\n" );
document.write( "this means that:<BR>\n" );
document.write( "y = 3-5 = -2<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "y = - (3-5)= -(-2) = 2<BR>\n" );
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document.write( "if y = -2, then 5 + |-2|  = 5 + 2 = 7 which does NOT satisfy the original equation of x + |y| = 3.<BR>\n" );
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document.write( "if y = 2, then 5 + |2| = 5 + 2 = 7 which does NOT satisfy the original equation of x + |y| = 3<BR>\n" );
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document.write( "bottom line is:<BR>\n" );
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document.write( "the domain is x <= 3<BR>\n" );
document.write( "the range is y can be any real number<BR>\n" );
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document.write( "a graph of the equation would show this to be true but you have to look at the graph only for the values of x <= 3.<BR>\n" );
document.write( "you have to ignore any values of x > 3.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%281200%2C300%2C-7%2C3%2C-10%2C10%2Cx-3%2C-%28x-3%29%29 ALIGN=MIDDLE  ALT=\"graph+%281200%2C300%2C-7%2C3%2C-10%2C10%2Cx-3%2C-%28x-3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "from this graph you can see that:<BR>\n" );
document.write( "when x = 3, y = 0<BR>\n" );
document.write( "when x = 0, y = +/- 3<BR>\n" );
document.write( "when x = -6, y = +/- 9<BR>\n" );
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document.write( "in all cases the |y| is positive and the equation of x + |y| = 3 is satisfied as long as x <= 3<BR>\n" );
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document.write( "PROBLEM NUMBER 2<BR>\n" );
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document.write( "domain and range of the function y = 2x^2-3?<BR>\n" );
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document.write( "looks like any real number of x will satisfy this equation.<BR>\n" );
document.write( "if x is a positive number, then y will be a positive number >= -3<BR>\n" );
document.write( "if x is 0 y will be = to -3<BR>\n" );
document.write( "if x is a negative number, then y will be a positive number >= -3<BR>\n" );
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document.write( "looks like the lowest y can be is -3.<BR>\n" );
document.write( "this happens when x = 0<BR>\n" );
document.write( "any number of x > 0 or < 0 will result in a value of y that is greater than -3.<BR>\n" );
document.write( "so:<BR>\n" );
document.write( "domain is all real numbers.<BR>\n" );
document.write( "range is all real numbers >= to -3<BR>\n" );
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document.write( "graph of this equation is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28+300%2C300%2C-5%2C5%2C-5%2C5%2Cx%5E2-3%29 ALIGN=MIDDLE  ALT=\"graph+%28+300%2C300%2C-5%2C5%2C-5%2C5%2Cx%5E2-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( " </SPAN>\n" );
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