Consider the function \(f(x)=\dfrac{6}{x}\).
- State the equation(s) of any asymptotes of \(f(x)\). (1 mark)
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- Complete the table of values of \(y=f(x)\) below. (1 mark)
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- \begin{array}{|c|c|c|c|c|c|c|}
\hline \rule{0pt}{2.5ex} \ \ \ x \ \ \ \rule[-1ex]{0pt}{0pt}& -\ 3 \ & & \quad 1 \quad & \ \ \ 3 \ \ \ \\
\hline \rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} & & -6 & & 2 \\
\hline
\end{array} - Sketch the graph of \(y=f(x)\), clearly showing any asymptote(s) and points from the table. (2 marks)
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a. \(x=0, \ y=0\)
b. \(\text{Table of values:}\)
\begin{array}{|c|c|c|c|c|c|c|}
\hline \rule{0pt}{2.5ex} \ \ \ x \ \ \ \rule[-1ex]{0pt}{0pt}& -\ 3 \ & \ -1 \ & \quad 1 \quad & \ \ \ 3 \ \ \ \\
\hline \rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} & -2 & -6 & 6 & 2 \\
\hline
\end{array}
c.
a. \(y=\dfrac{6}{x} \ \Rightarrow \ x \neq 0\)
\(\text{Vertical asymptote at} \ \ x=0.\)
\(\text{As}\ x \rightarrow \infty, \ y \rightarrow 0^{+}\)
\(\text{As}\ x \rightarrow -\infty, \ y \rightarrow 0^{-}\)
\(\text{Horizontal asymptote at} \ \ y=0.\)
b. \(\text{Table of values:}\)
\begin{array}{|c|c|c|c|c|c|c|}
\hline \rule{0pt}{2.5ex} \ \ \ x \ \ \ \rule[-1ex]{0pt}{0pt}& -\ 3 \ & \ -1 \ & \quad 1 \quad & \ \ \ 3 \ \ \ \\
\hline \rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} & -2 & -6 & 6 & 2 \\
\hline
\end{array}
c.