document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165857>Question 1165857</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the function f(x)=1/x\r<BR>\n" );
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document.write( "a) Write the equation g(x) after the following transformations:<BR>\n" );
document.write( "horizontal compression by the factor 1/5 ,<BR>\n" );
document.write( "vertical stretch by the factor 7<BR>\n" );
document.write( "reflection in the y-axis<BR>\n" );
document.write( "translation 10 units left and 1 unit down\r<BR>\n" );
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document.write( "b) State the domain and range of g(x)\r<BR>\n" );
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document.write( "Steps needed also </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852882 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=852882\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to determine the equation, domain, and range of $g(x)$ after the given transformations.\r<BR>\n" );
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document.write( "The original function is $f(x) = \frac{1}{x}$.\r<BR>\n" );
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document.write( "## a. Write the Equation $g(x)$\r<BR>\n" );
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document.write( "A transformation of a function $f(x)$ can be expressed in the form $g(x) = a \cdot f(b(x-h)) + k$, where:\r<BR>\n" );
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document.write( "* $|a|$ is the **vertical stretch/compression** factor. (A negative $a$ implies a reflection across the x-axis.)<BR>\n" );
document.write( "* $|1/b|$ is the **horizontal stretch/compression** factor. (A negative $b$ implies a reflection across the y-axis.)<BR>\n" );
document.write( "* $h$ is the **horizontal shift** (translation).<BR>\n" );
document.write( "* $k$ is the **vertical shift** (translation).\r<BR>\n" );
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document.write( "### Step 1: Apply Horizontal Compression and Reflection in the $y$-axis\r<BR>\n" );
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document.write( "* **Horizontal compression by factor $1/5$** means $b = 5$.<BR>\n" );
document.write( "* **Reflection in the $y$-axis** means $b$ is negative, so $b = -5$.\r<BR>\n" );
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document.write( "$$f(x) \to f(-5x) = \frac{1}{-5x}$$\r<BR>\n" );
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document.write( "### Step 2: Apply Vertical Stretch\r<BR>\n" );
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document.write( "* **Vertical stretch by factor 7** means $a = 7$.\r<BR>\n" );
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document.write( "$$f(-5x) \to 7f(-5x) = 7 \left(\frac{1}{-5x}\right) = -\frac{7}{5x}$$\r<BR>\n" );
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document.write( "### Step 3: Apply Translations\r<BR>\n" );
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document.write( "* **Translation 10 units left** means $h = -10$. We replace $x$ with $(x - (-10)) = (x+10)$.<BR>\n" );
document.write( "* **Translation 1 unit down** means $k = -1$.\r<BR>\n" );
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document.write( "Substitute $(x+10)$ for $x$ and add $k=-1$:<BR>\n" );
document.write( "$$g(x) = -\frac{7}{5(x+10)} - 1$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## The Final Equation $g(x)$\r<BR>\n" );
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document.write( "$$g(x) = -\frac{7}{5(x+10)} - 1$$\r<BR>\n" );
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document.write( "***\r<BR>\n" );
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document.write( "## b. State the Domain and Range of $g(x)$\r<BR>\n" );
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document.write( "The domain and range of a transformed function $g(x) = a \cdot f(b(x-h)) + k$ relate directly to the transformations applied to the parent function $f(x)$.\r<BR>\n" );
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document.write( "The parent function $f(x) = \frac{1}{x}$ has:<BR>\n" );
document.write( "* Domain: $x \neq 0$ (Vertical Asymptote at $x=0$)<BR>\n" );
document.write( "* Range: $y \neq 0$ (Horizontal Asymptote at $y=0$)\r<BR>\n" );
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document.write( "### Domain of $g(x)$\r<BR>\n" );
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document.write( "The domain is restricted by the **vertical asymptote**, which is shifted horizontally.<BR>\n" );
document.write( "The vertical asymptote of $f(x)$ is $x=0$.<BR>\n" );
document.write( "The only horizontal shift is **10 units left** ($h=-10$).<BR>\n" );
document.write( "* New Vertical Asymptote: $x = 0 - 10 = -10$.<BR>\n" );
document.write( "* The domain is all real numbers except $x=-10$.\r<BR>\n" );
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document.write( "$$\text{Domain of } g(x): \mathbf{\{x \in \mathbb{R} \mid x \neq -10\} \text{ or } (-\infty, -10) \cup (-10, \infty)}$$\r<BR>\n" );
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document.write( "### Range of $g(x)$\r<BR>\n" );
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document.write( "The range is restricted by the **horizontal asymptote**, which is shifted vertically.<BR>\n" );
document.write( "The horizontal asymptote of $f(x)$ is $y=0$.<BR>\n" );
document.write( "The only vertical shift is **1 unit down** ($k=-1$).<BR>\n" );
document.write( "* New Horizontal Asymptote: $y = 0 - 1 = -1$.<BR>\n" );
document.write( "* The range is all real numbers except $y=-1$.\r<BR>\n" );
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document.write( "$$\text{Range of } g(x): \mathbf{\{y \in \mathbb{R} \mid y \neq -1\} \text{ or } (-\infty, -1) \cup (-1, \infty)}$$ </SPAN>\n" );
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