document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171467>Question 1171467</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What can be said about the domain of the function  f \circ g  where  f(y)= \frac{4}{y-2}  and  g(x)= \frac{5}{3x-1} ? Express it in terms of a union of intervals of real numbers. Go to www.desmos.com/calculator and obtain the graph of  f ,  g , and  f \circ g .\r<BR>\n" );
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document.write( "Find the inverse of the function  f(x)=4+ \sqrt{x-2}  .<BR>\n" );
document.write( "State the domains and ranges of both the function and the inverse function in terms of intervals of real numbers.\r<BR>\n" );
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document.write( "Go to www.desmos.com/calculator and obtain the graph of  f , its inverse, and  g(x)=x  in the same system of axes. About what pair (a, a) are (11, 7) and (7, 11) reflected about? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850916 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=850916\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This question is identical to a previous question. Here's the solution again:\r<BR>\n" );
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document.write( "**1. Domain of f(g(x)) (f \circ g)**\r<BR>\n" );
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document.write( "* **f(y) = 4/(y - 2)**<BR>\n" );
document.write( "    * The domain of f(y) is all real numbers except y = 2.<BR>\n" );
document.write( "* **g(x) = 5/(3x - 1)**<BR>\n" );
document.write( "    * The domain of g(x) is all real numbers except 3x - 1 = 0, which means x = 1/3.<BR>\n" );
document.write( "* **f(g(x)) = f(5/(3x - 1)) = 4 / [(5/(3x - 1)) - 2]**<BR>\n" );
document.write( "    * To find the domain of f(g(x)), we need to consider two things:<BR>\n" );
document.write( "        * The domain of g(x) (x ≠ 1/3).<BR>\n" );
document.write( "        * The values of x that make the denominator of f(g(x)) equal to zero.<BR>\n" );
document.write( "    * Set the denominator to zero:<BR>\n" );
document.write( "        * 5/(3x - 1) - 2 = 0<BR>\n" );
document.write( "        * 5/(3x - 1) = 2<BR>\n" );
document.write( "        * 5 = 2(3x - 1)<BR>\n" );
document.write( "        * 5 = 6x - 2<BR>\n" );
document.write( "        * 7 = 6x<BR>\n" );
document.write( "        * x = 7/6<BR>\n" );
document.write( "    * Therefore, the domain of f(g(x)) is all real numbers except x = 1/3 and x = 7/6.<BR>\n" );
document.write( "    * In interval notation: (-∞, 1/3) U (1/3, 7/6) U (7/6, ∞)\r<BR>\n" );
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document.write( "* **Desmos Graphing:**<BR>\n" );
document.write( "    * Go to [www.desmos.com/calculator](https://www.desmos.com/calculator).<BR>\n" );
document.write( "    * Input:<BR>\n" );
document.write( "        * f(y) = 4/(y - 2)<BR>\n" );
document.write( "        * g(x) = 5/(3x - 1)<BR>\n" );
document.write( "        * f(g(x))<BR>\n" );
document.write( "    * Examine the graph of f(g(x)) to confirm the domain. You will see vertical asymptotes at x=1/3 and x=7/6.\r<BR>\n" );
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document.write( "**2. Inverse of f(x) = 4 + √(x - 2)**\r<BR>\n" );
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document.write( "* **Finding the inverse:**<BR>\n" );
document.write( "    * Let y = 4 + √(x - 2)<BR>\n" );
document.write( "    * Swap x and y: x = 4 + √(y - 2)<BR>\n" );
document.write( "    * Solve for y:<BR>\n" );
document.write( "        * x - 4 = √(y - 2)<BR>\n" );
document.write( "        * (x - 4)² = y - 2<BR>\n" );
document.write( "        * y = (x - 4)² + 2<BR>\n" );
document.write( "    * Therefore, f⁻¹(x) = (x - 4)² + 2\r<BR>\n" );
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document.write( "* **Domains and Ranges:**<BR>\n" );
document.write( "    * **f(x) = 4 + √(x - 2)**<BR>\n" );
document.write( "        * Domain: x - 2 ≥ 0 => x ≥ 2, or [2, ∞)<BR>\n" );
document.write( "        * Range: Since √(x - 2) ≥ 0, f(x) ≥ 4, or [4, ∞)<BR>\n" );
document.write( "    * **f⁻¹(x) = (x - 4)² + 2**<BR>\n" );
document.write( "        * Domain: The range of f(x) becomes the domain of f⁻¹(x), so x ≥ 4, or [4, ∞)<BR>\n" );
document.write( "        * Range: The domain of f(x) becomes the range of f⁻¹(x), so y ≥ 2, or [2, ∞)\r<BR>\n" );
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document.write( "* **Desmos Graphing:**<BR>\n" );
document.write( "    * Go to [www.desmos.com/calculator](https://www.desmos.com/calculator).<BR>\n" );
document.write( "    * Input:<BR>\n" );
document.write( "        * f(x) = 4 + sqrt(x - 2)<BR>\n" );
document.write( "        * y = (x - 4)² + 2 {x>=4}<BR>\n" );
document.write( "        * y = x<BR>\n" );
document.write( "    * The graph of f⁻¹(x) is the reflection of f(x) across the line y = x.<BR>\n" );
document.write( "    * The points (11, 7) and (7, 11) are reflections of each other across the line y = x.<BR>\n" );
document.write( "    * The point that is the center of the reflection is the intersection of the line y=x, and the line that goes through (11,7) and (7,11). The midpoint of (7,11) and (11,7) is (9,9). Therefore, the points are reflected about the pair (9, 9). </SPAN>\n" );
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