document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165630>Question 1165630</A>: <I><SPAN STYLE=\"word-break: break-all;\"> g={(-4,-9),(-2,8),(4,5),(8,4)}\r<BR>\n" );
document.write( "\n" );
document.write( "h(x)=2x-13\r<BR>\n" );
document.write( "\n" );
document.write( "Find the following<BR>\n" );
document.write( "g^-1(8)=<BR>\n" );
document.write( "h^-1(x)=<BR>\n" );
document.write( "(h∘h^-1)(-5)= </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852955 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.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=852955\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem involves finding inverse functions and evaluating a function composition.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## 1. Finding $g^{-1}(8)$\r<BR>\n" );
document.write( "\n" );
document.write( "The function $g$ is defined by the set of ordered pairs:<BR>\n" );
document.write( "$$g = \{(-4,-9), (-2,8), (4,5), (8,4)\}$$\r<BR>\n" );
document.write( "\n" );
document.write( "The inverse function $g^{-1}$ reverses the ordered pairs. To find $g^{-1}(8)$, we look for the pair in $g$ where the output (y-value) is 8.\r<BR>\n" );
document.write( "\n" );
document.write( "The pair in $g$ with an output of 8 is **$(-2, 8)$**.\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the input (x-value) for $g^{-1}(8)$ is $-2$.\r<BR>\n" );
document.write( "\n" );
document.write( "$$\mathbf{g^{-1}(8) = -2}$$\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## 2. Finding $h^{-1}(x)$\r<BR>\n" );
document.write( "\n" );
document.write( "The function is $h(x) = 2x - 13$. To find the inverse, we set $y = h(x)$ and solve for $x$ in terms of $y$.\r<BR>\n" );
document.write( "\n" );
document.write( "1.  Set $y = h(x)$:<BR>\n" );
document.write( "    $$y = 2x - 13$$<BR>\n" );
document.write( "2.  Swap $x$ and $y$:<BR>\n" );
document.write( "    $$x = 2y - 13$$<BR>\n" );
document.write( "3.  Solve for $y$:<BR>\n" );
document.write( "    $$x + 13 = 2y$$<BR>\n" );
document.write( "    $$y = \frac{x + 13}{2}$$\r<BR>\n" );
document.write( "\n" );
document.write( "$$\mathbf{h^{-1}(x) = \frac{x + 13}{2}}$$\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## 3. Finding $(h \circ h^{-1})(-5)$\r<BR>\n" );
document.write( "\n" );
document.write( "The expression $(h \circ h^{-1})(-5)$ represents the composition of the function $h$ with its inverse $h^{-1}$, evaluated at $x=-5$.\r<BR>\n" );
document.write( "\n" );
document.write( "By the definition of an inverse function, when a function is composed with its inverse, the result is the identity function:<BR>\n" );
document.write( "$$(f \circ f^{-1})(x) = x$$\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, for $h(x)$:\r<BR>\n" );
document.write( "\n" );
document.write( "$$(h \circ h^{-1})(-5) = -5$$\r<BR>\n" );
document.write( "\n" );
document.write( "$$\mathbf{(h \circ h^{-1})(-5) = -5}$$ </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );