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Functions, EXT1 EQ-Bank 10

Consider the function  \(f(\theta)=\operatorname{cosec}\left(\frac{\pi}{2}-\theta\right)\)  for  \(0 \leqslant \theta \leqslant 2 \pi\).

  1. Sketch the graph of  \(y=\operatorname{cosec}\left(\frac{\pi}{2}-\theta\right)\),  showing all key features.   (2 marks)

     
     
  2. In set notation, state the range of \(\theta\).   (1 mark)

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a.     


 

b.   \(\text{Range} \ \ f(\theta):\ y \in(-\infty,-1] \cup[1, \infty)\)

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a.    \(y=\operatorname{cosec}\left(\frac{\pi}{2}-\theta\right)=\dfrac{1}{\sin \left(\frac{\pi}{2}-\theta\right)}=\dfrac{1}{\cos\, \theta}\)
 


 

b.   \(\text{Range} \ \ f(\theta):\ y \in(-\infty,-1] \cup[1, \infty)\)

Functions, EXT1 EQ-Bank 9

Consider the functions  \(f(x)=\tan x\)  and  \(g(x)=\cot x\).

  1. Explain why  \(\cot x \neq \dfrac{1}{\tan x}\)  for all values of \(x\).   (2 marks)

  2. On the same set of axes below, sketch  \(y=\tan x\)  and  \(y=\cot x\)  for  \(0<x<\pi\), identifying any points where the graphs intersect.   (2 marks)
     
     
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a.    \(\text{At}\ \  x=\dfrac{\pi}{2}:\)

\(\cot \dfrac{\pi}{2}=\dfrac{\cos \frac{\pi}{2}}{\sin \frac{\pi}{2}}=\dfrac{0}{1}=0 \ \Rightarrow \ \text{defined}\).

\(\tan \dfrac{\pi}{2}=\dfrac{\sin \frac{\pi}{2}}{\cos \frac{\pi}{2}}=\dfrac{1}{0} \Rightarrow \ \text{undefined}\).

\(\dfrac{1}{\tan \frac{\pi}{2}}\ \ \text{is therefore undefined}\).

\(\therefore \cot x \neq \dfrac{1}{\tan x} \ \ \text{for all values of }\ x\).
 

b.
       

Show Worked Solution

a.    \(\text{At}\ \  x=\dfrac{\pi}{2}:\)

\(\cot \dfrac{\pi}{2}=\dfrac{\cos \frac{\pi}{2}}{\sin \frac{\pi}{2}}=\dfrac{0}{1}=0 \ \Rightarrow \ \text{defined}\).

\(\tan \dfrac{\pi}{2}=\dfrac{\sin \frac{\pi}{2}}{\cos \frac{\pi}{2}}=\dfrac{1}{0} \Rightarrow \ \text{undefined}\).

\(\dfrac{1}{\tan \frac{\pi}{2}}\ \ \text{is therefore undefined}\).

\(\therefore \cot x \neq \dfrac{1}{\tan x} \ \ \text{for all values of }\ x\).
 

b.
       

Functions, EXT1 EQ-Bank 8

Consider the function  \(y=\operatorname{cosec}\,x\)  for  \(-\pi \leqslant x \leqslant \pi\).

  1. State the equations of all vertical asymptotes in the given domain.   (1 mark)

  2. Sketch the graph of  \(y=\operatorname{cosec} x\), showing all key features.   (2 marks)

     
     
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a.    \(y=\operatorname{cosec}\,x=\dfrac{1}{\sin x}\)

\(\text{Asymptotes when \(\ \sin x=0 \ \) in given domain.}\)

\(\therefore \ \text{Asymptotes at} \ \ x=-\pi, 0, \pi\)
 

b.
       

Show Worked Solution

a.    \(y=\operatorname{cosec}\,x=\dfrac{1}{\sin x}\)

\(\text{Asymptotes when \(\ \sin x=0 \ \) in given domain.}\)

\(\therefore \ \text{Asymptotes at} \ \ x=-\pi, 0, \pi\)
 

b.
       

Functions, EXT1 EQ-Bank 7

  1. Sketch the graph of  \(y=\sec x\)  for  \(0 \leqslant x \leqslant 2 \pi\).
  2. In your answer, identify all asymptotes and the coordinates of any maximum and minimum turning points.   (2 marks)

     

  3. Using set notation, state the domain and range of  \(y=\sec x\).   (1 mark)

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a.
   

b.    \(\text{Domain:} \ x \in\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right) \cup\left(\frac{3 \pi}{2}, 2 \pi\right]\)

\(\text{Range:} \ y \in(-\infty,-1] \cup[1, \infty)\)

Show Worked Solution

a.    \(\text{Draw}\ \ y=\cos\,x\ \ \text{to inform graph:}\)

 
   

\(\text{Minimum TPs:}\ (0,1), (2\pi, 1) \)

\(\text{Maximum TP:}\ (\pi, -1)\)
 

b.    \(\text{Domain:} \ x \in\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right) \cup\left(\frac{3 \pi}{2}, 2 \pi\right]\)

\(\text{Range:} \ y \in(-\infty,-1] \cup[1, \infty)\)

Functions, EXT1′ F1 2007 HSC 3a*

The diagram shows the graph of  \(y = f(x)\). The line  \(y = x\)  is an asymptote.

Draw separate one-third page sketches of the graphs of the following:

  1.   \(f(\abs{x})\).   (2 marks)

  2.    \(f(x)-x\).   (2 marks)

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i.       
       

ii.
           

Show Worked Solution
MARKER’S COMMENT: In part (i), a significant number of students graphed  \(y=\abs{f(x)}\).
i.

 

ii. 

Functions, EXT1′ F1 2019 HSC 12d

Consider the function  \(f(x) = x^3-1\).

  1.  Sketch the graph  \(y = \abs{f(x)}\).   (1 mark)

  2.  Sketch the graph  \(y = \dfrac{1}{f(x)}\).   (2 marks)

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i.    \(y = \abs{x^3-1}\)

ii.  \(y = \dfrac{1}{x^3-1}\)

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i.    \(y = \abs{x^3-1}\)

ii.  \(y = \dfrac{1}{x^3-1}\)

Functions, EXT1′ F1 2013 HSC 13bii

The diagram shows the graph of a function `f(x).`
 

Sketch the curve  `y = 1/(1-f(x)).`   (3 marks)

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Show Worked Solution

`y = 1/(1-f(x))`

MARKER’S COMMENT: Correct working sketches such as `y=-f(x)` and `y=1-f(x)` meant that students could obtain some marks, even if their final sketch was wrong.

`f(x) = 1,\ \ \ y\ text(undefined.)`

`f(x) > 1,\ \ \ y < 0`

`f(x) <= 0, \ \ \ y <= 1`

`\text{Create graph in 3 stages:}`
 

Functions, EXT1 F1 2024 HSC 6 MC

How many real value(s) of \(x\) satisfy the equation

\(\abs{b} = \abs{b\,\sin(4x)}\),

where  \(x \in [0, 2\pi]\) and \(b\) is not zero?

  1. \(1\)
  2. \(2\)
  3. \(4\)
  4. \(8\)
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\(D\)

Show Worked Solution

\(\abs{b} = \abs{b\,\sin(4x)}\ \ \Rightarrow\ \ \abs{1} = \abs{\sin(4x)}\ \ (b \neq 0)\)

\(\sin(4x) = \pm 1\)

\(x \in [0, 2\pi]\ \ \Rightarrow\ \ 4x \in [0, 8\pi]\)

\(\therefore 8\ \text{real solutions}\)

\(\Rightarrow D\)

♦ Mean mark 53%.

Functions, EXT1 F1 EQ-Bank 12

The diagram shows the graph of the function \(y=f(x)\).
 

  

On the diagram, sketch the graph of  \(y=\dfrac{1}{f(x)}\), stating the range in set notation.   (3 marks)

 

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\(\text{Range}\ y=\dfrac{1}{f(x)}:  \Big(-\infty, -2\Big)\ \cup\  \Big[\dfrac{2}{3}, \infty\Big)\)

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Note: graphs must intersect on \(y=1\) line.

\(\text{Range}\ y=\dfrac{1}{f(x)}:  \Big(-\infty, -2\Big)\ \cup\  \Big[\dfrac{2}{3}, \infty\Big)\)

Functions, EXT1 F1 EQ-Bank 2

Sketch the graph of  \(y=\abs{x^2-5x+4}\).   (2 marks)

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Show Worked Solution

\(y=\abs{x^2-5 x+4}=\abs{(x-1)(x-4)}\)

\(x\text{-intercepts at} \ (1,0), (4,0)\)

\(y=(x-1)(x-4) \ \text{has low at} \ \ x=\dfrac{5}{2} \ \text{(by symmetry)}\)

\(\Rightarrow \ \text {Low at} \ \left(\dfrac{5}{2},\dfrac{-9}{4}\right)\)
 

Functions, EXT1 F1 2023 HSC 8 MC

The diagram shows the graph of a function.
 

Which of the following is the equation of the function?

  1. \(y=\Big{|}1-\big{|}|x|-2\big{|}\Big{|}\)
  2. \(y=\Big{|}2-\big{|}|x|-1\big{|}\Big{|}\)
  3. \(y=\Big{|}1-\big{|}x-2\big{|}\Big{|}\)
  4. \(y=\Big{|}2-\big{|}x-1\big{|}\Big{|}\)
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\(A\)

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\(\text{By elimination:}\)

\(\text{Even function}\ \rightarrow\ \text{Eliminate}\ C\ \text{and}\ D \)

\(\text{Graph passes through}\ (1, 0) \)

\(\text{Option}\ A:\ \ y=\Big{|}1-\big{|}1-2\big{|}\Big{|} =\Big{|}1-1\Big{|}=0\ \ \text{(lies on graph)} \)

\(\text{Option}\ B:\ \ y=\Big{|}2-\big{|}|1|-1\big{|}\Big{|} =\Big{|}2-0\Big{|}=2\ \ \text{(not on graph)} \)

\(\therefore\ \text{Eliminate}\ B \)

\(\Rightarrow A\)

Functions, EXT1 F1 2022 HSC 4 MC

The diagram shows the graph of the sum of the functions `f(x)` and `g(x)`.
 

Which of the following best represents the graphs of both `f(x)` and `g(x)`?
 


 

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`A`

Show Worked Solution

`text{By Elimination,}`

`text{Consider the}\ ytext{-axis intercept of both graphs in each option:}`

`B and C\ text{will have a positive “net”}\ ytext{-value intercept (Eliminate}\ B\ \text{and}\ C).`
 

`y=f(x)+g(x)=0\ \ text{when the two graphs are equidistant from the}`

`xtext{-axis (Eliminate}\ D).`

`=>A`

Functions, EXT1 F1 2021 HSC 7 MC

Which curve best represents the graph of the function  `f(x) = -a sin x + b cos x`  given that the constants `a` and `b` are both positive?
 

A. B.
C. D.
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`D`

Show Worked Solution

`text(By elimination:)`

`text(At)\ \ x = 0, -asinx = 0\ \ text(and)\ \ bcosx = b`

COMMENT: When told constants are “positive”, pay close attention!

`:. f(0) = b > 0`

`->\ text(Eliminate A and C)`
 

`f^{′}(x) = -acosx-bsinx`

`text(At)\ \ x = 0, -acos x = -a\ \ text(and)\ \ bsinx = 0`

`:. f′(0) = -a < 0`

`->\ text(Eliminate B)`
 

`=>\ D`

Functions, EXT1′ F1 2008 HSC 3a

The following diagram shows the graph of  `y = g(x)`.
 

 
Draw separate one-third page sketches of the graphs of the following:

  1.  `y = |g(x)|`  (1 mark)

  2.  `y = 1/(g(x))`  (2 marks)

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  1.  
  2.  
Show Worked Solution
i.   

 

ii.   

Functions, EXT1 F1 SM-Bank 1

The diagram shows the graph of the function  `f(x) = x/(x-1)`.
  


 

If  `g(x)=x`, draw the graph of  `y = f(x) + g(x)`   (2 marks)

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Show Worked Solution
  `y` `= x + x/(x-1)`
    `= x + ((x-1) + 1)/(x-1)`
    `= x + 1 + 1/(x-1)`

 
`text(As)\ x -> ∞, \ y -> x + 1`

`text(As)\ x ->-∞, \ y -> x + 1`
 
 

Functions, EXT1 F1 SM-Bank 12

Given  `f(x) = x^3 - x^2 - 2x`, without calculus sketch a separate half page graph of the following functions, showing all asymptotes and intercepts.

  1.   `y = |\ f(x)\ |`  (1 mark)

  2.   `y = f(|x|)`  (2 marks)

  3.    `y = 1/(f(x))`  (2 marks)

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  1.  
  2.   
  3.   
Show Worked Solution
i.    `f(x)` `= x^3 – x^2 – 2x`
    `= x(x^2 – x – 2)`
    `= x(x – 2)(x + 1)`

 

 

ii.   

 
`y = f(|x|)\ text(is a reflection of)\ y = f(x)\ text(for)\ x > 0`

`text(is the)\ ytext(-axis.)`

 

iii.

Functions, EXT1 F1 2017 HSC 12b

  1. Carefully sketch the graphs of  \(y = \abs{x + 1}\)  and  \(y = 3-\abs{x-2}\)  on the same axes, showing all intercepts.   (3 marks)

  2. Using the graphs from part (i), or otherwise, find the range of values of \(x\) for which
  3. \(\abs{x + 1} + \abs{x-2}= 3\).   (1 mark)

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 i.    

ii.   \(-1 \leqslant x \leqslant 2\)

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i.   

 

ii.    \(\abs{x + 1} + \abs{x-2}\) \(= 3\)
  \(\abs{x + 1}\) \(= 3-\abs{x-2}\)

 
\(\therefore\ \text{Solution is intersection of the graphs:}\)

\(-1 \leqslant x \leqslant 2\)

Functions, EXT1′ F1 2010 HSC 3a

  1.  Sketch the graph  `y = x^2 + 4x`.   (1 mark)

  2.  Sketch the graph  `y = 1/(x^2 + 4x)`.   (2 marks)

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

Graphs, EXT2 2010 HSC 3ai1

 

ii.  

Graphs, EXT2 2010 HSC 3aii 

Functions, EXT1′ F1 2014 HSC 12a

The diagram shows the graph of a function  `f(x)`.
 

Graphs, EXT2 2014 HSC 12a
 

Draw a separate half-page graph for each of the following, showing all asymptotes and intercepts.

  1.   `y = f(|\ x\ |)`  (2 marks)

  2.   `y = 1/(f(x))`  (2 marks)

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

Graphs, EXT2 2014 HSC 12a Answer1

 

ii.  

Graphs, EXT2 2014 HSC 12a Answer2

Functions, EXT1 F1 2008 HSC 3a

  1.  Sketch the graph of  `y = |\ 2x - 1\ |`.   (1 mark)

  2.  Hence, or otherwise, solve  `|\ 2x - 1\ | <= |\ x - 3\ |`.    (3 marks)

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  1.  
    Real Functions, EXT1 2008 HSC 3a Answer

  2. `-2 <= x <= 4/3`
Show Worked Solution
i.    Real Functions, EXT1 2008 HSC 3a Answer

 

ii.  `text(Solving for)\ \ |\ 2x – 1\ | <= |\ x – 3\ |`

`text(Graph shows the statement is TRUE)`

`text(between the points of intersection.)`
 

`=>\ text(Intersection occurs when)`

`(2x – 1)` `= (x – 3)\ \ \ text(or)\ \ \ ` `-(2x – 1)` `= x – 3`
`x` `= -2` `-2x + 1` `= x – 3`
    `-3x` `= -4`
    `x` `= 4/3`

 

`:.\ text(Solution is)\ \ {x: -2 <=  x <= 4/3}`