document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=402714>Question 402714</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help with the following:\r<BR>\n" );
document.write( "\n" );
document.write( "5. Is f(x) = x^2+2x-3 a one-to-one function? Explain your answer \r<BR>\n" );
document.write( "\n" );
document.write( "6. For the polynomial f(x)= -11x^3+3x^2-x+1, what is the degree of F and what is its leading coefficent? <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 #284880 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=284880\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1.\r<BR>\n" );
document.write( "\n" );
document.write( "One-to-One Function\r<BR>\n" );
document.write( "\n" );
document.write( "A function for which <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=every ALIGN=MIDDLE  ALT=\"every\"   DISABLED_style= \"max-width:90%; height:auto;\" > element of the range of the function corresponds to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=exactly ALIGN=MIDDLE  ALT=\"exactly\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+one ALIGN=MIDDLE  ALT=\"+one\"   DISABLED_style= \"max-width:90%; height:auto;\" > element of the domain. One-to-one is often written 1-1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " Vertical Line Test\r<BR>\n" );
document.write( "\n" );
document.write( "A test use to determine if a relation is a function. A relation is a function if there are no vertical lines that intersect the graph at more than one point.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Horizontal Line Test\r<BR>\n" );
document.write( "\n" );
document.write( "A test use to determine if a function is one-to-one. If a horizontal line intersects a function's graph more than once, then the function is not one-to-one.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: y = f(x) is a function if it passes the vertical line test. It is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-1+ ALIGN=MIDDLE  ALT=\"1-1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=if ALIGN=MIDDLE  ALT=\"if\"   DISABLED_style= \"max-width:90%; height:auto;\" > it passes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=both ALIGN=MIDDLE  ALT=\"both\"   DISABLED_style= \"max-width:90%; height:auto;\" > the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=vertical ALIGN=MIDDLE  ALT=\"vertical\"   DISABLED_style= \"max-width:90%; height:auto;\" > line test and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=horizontal ALIGN=MIDDLE  ALT=\"horizontal\"   DISABLED_style= \"max-width:90%; height:auto;\" > line test. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" > from A to B is called <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=one-to-one ALIGN=MIDDLE  ALT=\"one-to-one\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or 1-1) if whenever<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f+%28a%29+=+f+%28b%29 ALIGN=MIDDLE  ALT=\"f+%28a%29+=+f+%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b ALIGN=MIDDLE  ALT=\"a+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >.   No element of B is the image of more than one element in A.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In a one-to-one function, given any y there is only one x that can be paired with the given y.  Such functions are referred to as injective.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A function \"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" > \" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=HAS ALIGN=MIDDLE  ALT=\"HAS\"   DISABLED_style= \"max-width:90%; height:auto;\" > an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=INVERSE ALIGN=MIDDLE  ALT=\"INVERSE\"   DISABLED_style= \"max-width:90%; height:auto;\" > function \" f \" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=if ALIGN=MIDDLE  ALT=\"if\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=only ALIGN=MIDDLE  ALT=\"only\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+if ALIGN=MIDDLE  ALT=\"+if\"   DISABLED_style= \"max-width:90%; height:auto;\" > \" f \" is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=one-to-one ALIGN=MIDDLE  ALT=\"one-to-one\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "so, check if your function has an inverse:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=Plot_Inverse_Function.solver>PLOT a FUNCTION and its INVERSE</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Plotting a graph:<BR><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph_and_inverse%28+600%2C+400%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2%2B2x-3+%29 ALIGN=MIDDLE  ALT=\"graph_and_inverse%28+600%2C+400%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2%2B2x-3+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "\n" );
document.write( "  The function is shown in <FONT COLOR=RED><B>RED</B></FONT> and the inverse is snown in <FONT COLOR=GREEN><B>GREEN</B></FONT>. <BR>\n" );
document.write( "  This solver uses <A  rel=nofollow HREF=/services/rendering/>formula rendering system</A> and here's the actual formula that was plotted:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  graph_and_inverse( 600, 400, -10, 10, -10, 10, x^2+2x-3 )<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  </TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "it does have an inverse, thus it is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=one-to-one ALIGN=MIDDLE  ALT=\"one-to-one\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=+-11x%5E3%2B3x%5E2-x%2B1 ALIGN=MIDDLE  ALT=\"f%28x%29=+-11x%5E3%2B3x%5E2-x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, what is the degree of F and what is its leading coefficient? \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1.\r<BR>\n" );
document.write( "\n" );
document.write( "The coefficient of the term with the highest degree (greatest power of x) is called the leading coefficient and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cannot ALIGN=MIDDLE  ALT=\"cannot\"   DISABLED_style= \"max-width:90%; height:auto;\" > equal 0.\r<BR>\n" );
document.write( "\n" );
document.write( "A polynomial can have any non-negative degree. In other words, the polynomial's degree must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3E=+0 ALIGN=MIDDLE  ALT=\"n%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "2.The leading coefficient of a polynomial is the coefficient that is associated with the variable with the highest degree. \r<BR>\n" );
document.write( "\n" );
document.write( "your  polynomial is a polynomial of degree <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > with leading coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-11 ALIGN=MIDDLE  ALT=\"-11\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );