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Functions, 2ADV EQ-Bank 25

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 21

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 11

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 28

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`

Functions, 2ADV EQ-Bank 24

  1. Graph  \(y=4\)  and  \(y=|2 x+2|\) on the same number plane.   (2 marks)
     


  2. Hence, solve  \(|2 x+2|=4\).   (1 mark)

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a.
     
 

b.    \(\text{Solutions occur at graph intersections:}\)

\(\Rightarrow x=1 \ \ \text{or} \ \ x=-3.\)

Show Worked Solution

a.
     
 

b.    \(\text{Solutions occur at graph intersections:}\)

\(\Rightarrow x=1 \ \ \text{or} \ \ x=-3.\)

Functions, 2ADV EQ-Bank 13

A set of ordered pairs \((x, y)\) on the coordinate plane are represented by set \(A\) below:

\(A=\{(1,3),(4,6),(5,6),(0,1),(1,7)\}\)

Explain if Set \(A\) a function or a relation?   (2 marks)

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\(\text {If set \(A\) is a function, there can only be one \(y\)-value}\)

\(\text{for any given } x \text{-value.}\)

\(\text{This is not the case} \ \Rightarrow\ (1,3),(1,7)\)

\(\therefore\ \text{Set} \ A \ \text {is a relation.}\)

Show Worked Solution

\(\text {If set \(A\) is a function, there can only be one \(y\)-value}\)

\(\text{for any given } x \text{-value.}\)

\(\text{This is not the case} \ \Rightarrow\ (1,3),(1,7)\)

\(\therefore\ \text{Set} \ A \ \text {is a relation.}\)

Functions, 2ADV F1 2025 HSC 3 MC

What is the domain of the function  \(y=\sqrt{6-x^2}\) ?

  1. \(\left(0, \sqrt{6}\right)\)
  2. \(\left[0, \sqrt{6}\right]\)
  3. \(\left(-\sqrt{6}, \sqrt{6}\right)\)
  4. \(\left[-\sqrt{6}, \sqrt{6}\right]\)
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\(D\)

Show Worked Solution

\(6-x^2 \geq 6\ \ \Rightarrow\ \ x^2 \leq 6\)

\(-\sqrt{6} \leq x \leq \sqrt{6} \)

\(\Rightarrow D\)

Functions, 2ADV F1 2024 HSC 6 MC

What is the domain of the function  \(f(x)=\dfrac{1}{\sqrt{x^2-1}}\) ?

  1. \([-1,1]\)
  2. \((-\infty,-1] \cup[1, \infty)\)
  3. \((-1,1)\)
  4. \((-\infty,-1) \cup(1, \infty)\)
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\( D \)

Show Worked Solution
\(x^2-1\) \(>0\)  
\(x^2\) \(>1\)  

 
\( x<-1 \ \cup \ x>1 \)

\( x \in(-\infty,-1) \cup(1, \infty) \)

\( \Rightarrow D \)

Functions, 2ADV F1 2020 HSC 1 MC

Which inequality gives the domain of  `y = sqrt(2x-3)`?

  1. `x < 3/2`
  2. `x > 3/2`
  3. `x <= 3/2`
  4. `x >= 3/2`
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`D`

Show Worked Solution

`text(Domain exists when:)`

`2x-3` `>= 0`
`2x` `>= 3`
`x` `>= 3/2`

  
`=>D`

Functions, 2ADV F1 EQ-Bank 16

Find the values of `x` for which  `|x-3| = 1`.   (2 marks)

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`x=2 or 4`

Show Worked Solution

`|x-3| = 1`

`text(Method 1)`

`(x-3)^2` `=1`  
`x^2-6x +8` `=0`  
`(x-4)(x-2)` `=0`  

 
`:. x=2 or 4`
 

`text(Method 2)`

`(x-3)` `=1` `-(x-3)` ` = 1`
`x` `=4` `-x +3` `=1`
    `x` `=2`

Functions, 2ADV F1 EQ-Bank 19

Find all values of `x` for which  `| x-4 | = x/2 + 7`.   (3 marks)

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`x = 22\ \ text(or)\ -2`

Show Worked Solution
`x-4` `= x/2 + 7` `text(or)` `-(x-4)` `= x/2 + 7`
`2x-8` `= x + 14`   `-2x + 8` `= x + 14`
`x` `= 22`   `3x` `= -6`
      `x` `= -2`

 
`:. x = 22\ \ text(or)\ -2`

Functions, 2ADV F1 2019 HSC 13e

  1. Sketch the graph of  `y = |x-1|`  for  `-4 <= x <= 4`.   (1 mark)

  2. Using the sketch from part i, or otherwise, solve  `|x-1| = 2x + 4`.   (2 marks)

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  1. `text(See Worked Solutions)`
  2. `(-1, 2)`
Show Worked Solution
i.   

 

ii.    `text(By inspection, intersection when)\ x = -1`

`text(Test:)`

`|-1-1|` `= -2 + 4`
`2` `= 2`

 
`:.\ text(Intersection at)\ (-1, 2)`

Functions, 2ADV F1 EQ-Bank 17

  1.  State the domain and range of the function  `f(x) = -sqrt(12-x^2)`.   (2 marks)

  2.  Sketch the graph of `f(x)`.   (1 mark)

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a.    `text(Domain:)\ -sqrt12<=x<= sqrt12`

`text(Range:)\ -sqrt12<=y<= 0`

b.   

Show Worked Solution

a.   `y = -sqrt(12-x^2)`

`text(Domain:)\ -sqrt12<=x<= sqrt12`

`text(Range:)\ -sqrt12<=y<= 0`
 

b.  

Functions, 2ADV F1 2017 HSC 11h

Find the domain of the function  `f(x) = sqrt (3-x)`.  (2 marks)

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`x <= 3 or (-oo,3].`

Show Worked Solution

`text(Domain of)\ \ f(x) = sqrt (3-x)`

`3-x` `>= 0`
`x` `<= 3`

 

`text(Note domain can also be expressed as:)\ \ (-oo,3]`

Functions, 2ADV F1 2008 HSC 1d

Solve  `| 4x-3 | = 7`.   (2 marks)

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`x = 5/2\ \ text(or)\ x = -1`

Show Worked Solution

`| 4x-3 | = 7`

`4x-3` `= 7` `\ \ \ \ \ -(4x-3)` `= 7`
`4x` `= 10` `-4x + 3` `= 7`
`x` `= 5/2` `-4x` `= 4`
    `x` `= -1`

 

`:. x=5/2\ \ text(or)\ \ -1`

Functions, 2ADV F1 2010 HSC 1g

Let  `f(x) = sqrt(x-8)`.  What is the domain of  `f(x)`?   (1 mark)

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 `x >= 8`

Show Worked Solution
♦ Mean mark 49%.
MARKER’S COMMENT: `x>8` was a common incorrect answer.

`f(x) = sqrt(x-8)`

`text(Domain exists for:)`

`(x-8)` `>= 0`
`x` `>= 8`

Functions, 2ADV F1 2010 HSC 1d

Solve  `| 2x + 3 | = 9`.   (2 marks)

Show Answers Only

 `x = 3\ text(or)\ -6`

Show Worked Solution

`| 2x + 3 | = 9`

`2x + 3` `=9` `\ \ \ \ -(2x + 3)` `=9`
`2x` `=6` `\ \ \ \ -2x\ – 3` `=9`
`x` `=3` `\ \ \ \ -2x` `=12`
     `x` `=-6`

 

`:. x = 3\ \ text(or)\ \ -6`

Functions, 2ADV F1 2013 HSC 3 MC

Which inequality defines the domain of the function  `f(x) = 1/sqrt(x+3)` ?

  1. `x > -3`  
  2. `x >= -3`  
  3. `x < -3`  
  4. `x <= -3` 
Show Answers Only

`A`

Show Worked Solution

`text(Given)\ f(x) = 1/sqrt(x+3)`

`(x + 3)` `> 0`
`x` `> -3`

 

`:.\ text(The domain of)\ f(x)\ text(is)\ \ \ f(x)> -3`

`=>  A`