document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165124>Question 1165124</A>: <I><SPAN STYLE=\"word-break: break-all;\"> g={(-7,1)(-4,5)(1,-1)(7,3)}<BR>\n" );
document.write( "h(x)=4x-9<BR>\n" );
document.write( "Find the following.<BR>\n" );
document.write( "g^-1(1)= <BR>\n" );
document.write( "h^-1(x)=<BR>\n" );
document.write( "(h^-1∘h)(1)= </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853115 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=853115\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here are the solutions for the inverse function problems.\r<BR>\n" );
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document.write( "## 1. Find $g^{-1}(1)$\r<BR>\n" );
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document.write( "The function $g$ is defined by a set of ordered pairs:<BR>\n" );
document.write( "$$g = \{(-7, 1), (-4, 5), (1, -1), (7, 3)\}$$\r<BR>\n" );
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document.write( "The inverse function, $g^{-1}$, reverses the ordered pairs: if $(a, b)$ is in $g$, then $(b, a)$ is in $g^{-1}$.<BR>\n" );
document.write( "$$g^{-1} = \{(1, -7), (5, -4), (-1, 1), (3, 7)\}$$\r<BR>\n" );
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document.write( "To find $g^{-1}(1)$, look for the pair in $g^{-1}$ where the input (x-value) is 1.<BR>\n" );
document.write( "The pair is $(1, -7)$.\r<BR>\n" );
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document.write( "$$\mathbf{g^{-1}(1) = -7}$$\r<BR>\n" );
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document.write( "*Alternatively, we look for the output (y-value) of 1 in the original function $g$. Since $g(-7) = 1$, then $g^{-1}(1) = -7$.*\r<BR>\n" );
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document.write( "## 2. Find $h^{-1}(x)$\r<BR>\n" );
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document.write( "The function $h(x)$ is given by:<BR>\n" );
document.write( "$$h(x) = 4x - 9$$\r<BR>\n" );
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document.write( "To find the inverse function, $h^{-1}(x)$:<BR>\n" );
document.write( "1.  **Replace $h(x)$ with $y$**:<BR>\n" );
document.write( "    $$y = 4x - 9$$<BR>\n" );
document.write( "2.  **Swap $x$ and $y$**:<BR>\n" );
document.write( "    $$x = 4y - 9$$<BR>\n" );
document.write( "3.  **Solve for $y$**:<BR>\n" );
document.write( "    $$x + 9 = 4y$$<BR>\n" );
document.write( "    $$y = \frac{x + 9}{4}$$<BR>\n" );
document.write( "4.  **Replace $y$ with $h^{-1}(x)$**:\r<BR>\n" );
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document.write( "$$\mathbf{h^{-1}(x) = \frac{x + 9}{4}}$$\r<BR>\n" );
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document.write( "## 3. Find $(h^{-1} \circ h)(1)$\r<BR>\n" );
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document.write( "The composition $(h^{-1} \circ h)(x)$ is defined as $h^{-1}(h(x))$.\r<BR>\n" );
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document.write( "Since $h(x)$ and $h^{-1}(x)$ are inverses of each other, their composition always returns the original input, $x$, for any value in the domain.\r<BR>\n" );
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document.write( "$$(h^{-1} \circ h)(x) = x$$\r<BR>\n" );
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document.write( "Therefore, for the input $x=1$:<BR>\n" );
document.write( "$$(h^{-1} \circ h)(1) = 1$$\r<BR>\n" );
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document.write( "*Alternatively, solving step-by-step:*<BR>\n" );
document.write( "1.  **Find $h(1)$**:<BR>\n" );
document.write( "    $$h(1) = 4(1) - 9 = 4 - 9 = -5$$<BR>\n" );
document.write( "2.  **Find $h^{-1}(h(1))$, which is $h^{-1}(-5)$**:<BR>\n" );
document.write( "    $$h^{-1}(-5) = \frac{(-5) + 9}{4} = \frac{4}{4} = 1$$\r<BR>\n" );
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document.write( "$$\mathbf{(h^{-1} \circ h)(1) = 1}$$ </SPAN>\n" );
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