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Functions, 2ADV EQ-Bank 5

The cost per person \((C)\), in dollars, for hiring a function room varies inversely with the number of people \((n)\) attending, since the total venue cost is fixed at $2400.

  1. Write an equation relating \(C\) and \(n\).   (1 mark)

  2. Sketch a graph showing the relationship between the number of people and cost per person for  \(0<n \leqslant 100\). Label your axes.   (2 marks)
     

           

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a.    \(C=\dfrac{2400}{n}\)

b.    
         

Show Worked Solution

a.    \(C=\dfrac{2400}{n}\)

b.   \(\text{Table of values:}\)

\(\begin{array}{|c|c|c|c|c|}
\hline \rule{0pt}{2.5ex}\ \ n \ \ \rule[-1ex]{0pt}{0pt}& 20 & \ 40 \ & \ 60 \ & 100 \\
\hline \rule{0pt}{2.5ex}\ \ C \ \ \rule[-1ex]{0pt}{0pt}& 120 & 60 & 40 & 24 \\
\hline
\end{array}\)
 

Functions, 2ADV EQ-Bank 4

Consider the function  \(g(x)=-\dfrac{12}{x}\)

  1. Is \(g(x)\) an odd or even function? Give reason(s) for your answer.   (2 marks)

  2. Sketch the graph \(y=g(x)\) in the domain \(-6 \leq x \leq 6\). Label the endpoints and two other points on the curve.   (2 marks)

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a.    \(\text{If}\ g(x)\ \text{is odd}\ \ \Rightarrow \ \ g(-x)=-g(x)\)

\(-g(x)= \dfrac{12}{x}\)

\(g(-x) = -\dfrac{12}{(-x)} = \dfrac{12}{x} = -g(x)\)

\(\therefore g(x)\ \text{is odd (and cannot be even).}\)
 

b.    \(\text{Table of values:}\)

\begin{array}{|c|c|c|c|c|c|}
\hline \rule{0pt}{2.5ex}\ \ x \ \ \rule[-1ex]{0pt}{0pt}& -4 & -2 & \ \ 0 \ \ & 2 & 4 \\
\hline \rule{0pt}{2.5ex}y \rule[-1ex]{0pt}{0pt}& 3 & 6 & \infty & -6 & -3 \\
\hline
\end{array}

\(\text{Endpoints:}\ (-6,2), (6,-2) \)
 

Show Worked Solution

a.    \(\text{If}\ g(x)\ \text{is odd}\ \ \Rightarrow \ \ g(-x)=-g(x)\)

\(-g(x)= \dfrac{12}{x}\)

\(g(-x) = -\dfrac{12}{(-x)} = \dfrac{12}{x} = -g(x)\)

\(\therefore g(x)\ \text{is odd.}\)
 

b.    \(\text{Table of values:}\)

\begin{array}{|c|c|c|c|c|c|}
\hline \rule{0pt}{2.5ex}\ \ x \ \ \rule[-1ex]{0pt}{0pt}& -4 & -2 & \ \ 0 \ \ & 2 & 4 \\
\hline \rule{0pt}{2.5ex}y \rule[-1ex]{0pt}{0pt}& 3 & 6 & \infty & -6 & -3 \\
\hline
\end{array}

\(\text{Endpoints:}\ (-6,2), (6,-2) \)
 

Functions, 2ADV EQ-Bank 3

Consider the function  \(f(x)=\dfrac{6}{x}\).

  1. State the equation(s) of any asymptotes of \(f(x)\).   (1 mark)

  2. Complete the table of values of  \(y=f(x)\)  below.   (1 mark)

  3. \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}
  4. 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.   
         

Show Worked Solution

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.   
         

Functions, 2ADV EQ-Bank 7

The cost of hiring an open space for a music festival is  $120 000. The cost will be shared equally by the people attending the festival, so that `C` (in dollars) is the cost per person when `n` people attend the festival.

  1. Complete the table below and draw the graph showing the relationship between `n` and `C`.   (2 marks)
    \begin{array} {|l|c|c|c|c|c|c|}
    \hline
    \rule{0pt}{2.5ex}\text{Number of people} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
    \hline
    \rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} &  &  &  & 60 & 48\ & 40 \ \\
    \hline
    \end{array}

     

  2. What equation represents the relationship between `n` and `C`?   (1 mark)

  3. Give ONE limitation of this equation in relation to this context.   (1 mark)

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a.   

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of people} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
\hline
\rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} & 240 & 120 & 80 & 60 & 48\ & 40 \ \\
\hline
\end{array}

b.   `C = (120\ 000)/n` 

c.   `text(Limitations can include:)`

  `•\ n\ text(must be a whole number)`

  `•\ C > 0`

Show Worked Solution

a.   

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of people} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
\hline
\rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} & 240 & 120 & 80 & 60 & 48\ & 40 \ \\
\hline
\end{array}

b.   `C = (120\ 000)/n` 

c.   `text(Limitations can include:)`

  `•\ n\ text(must be a whole number)`

  `•\ C > 0`