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Functions, EXT1′ F1 2019 HSC 12d

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

  1.  Sketch the graph  `y = |\ f(x)\ |`.  (1 mark)

  2.  Sketch the graph  `y = 1/(f(x))`.  (2 marks)

  3.  Without using calculus, sketch the graph  `y = x/(f(x))`.  (2 marks)

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

i.   `y = |\ x^3 – 1\ |`

 

ii.   `y = 1/(x^3 – 1)`

 

iii.   `y = x/(x^3 – 1)`

Functions, EXT1′ F1 2018 HSC 12d

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


 

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

  1.  `y = |\ f(x)\ |`  (1 mark)

  2.  `y = (f(x))^2`  (2 marks)

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i.   

ii.

Show Worked Solution
i.   

 

ii.   

Functions, EXT1′ F1 2018 HSC 4 MC

Which graph best represents the curve  `y = 1/sqrt(x^2 - 4)`?
 

A.    B.   
C.    D.   
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`=>\ text(C)`

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`text(S)text(ince)\ \ sqrt(x^2 – 4) > 0`

`=> y > 0\ \ (text(Eliminate B and D))`
 

`text(As)\ x -> 2^+, sqrt(x^2 – 4) -> 0, y -> ∞`

`text(As)\ x -> -2^-, sqrt(x^2 – 4) -> 0, y -> ∞`

`=>\ text(C)`

Functions, EXT1′ F1 2017 HSC 8 MC

Suppose that  `f(x)`  is a non-zero odd function.

Which of the functions below is also odd?

  1. `f(x^2)cosx`
  2. `f(f(x))`
  3. `f(x^3)sinx`
  4. `f(x^2) - f(x)`
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`B`

Show Worked Solution

`text(By trial and error, consider)\ B:`

`f(f(−x))` `= f(−f(x))`
  `= −f(f(x))`

 
`:. f(f(x))\ text(is odd.)`

`=> B`

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(-x).`   (1 mark)

  2.   `f(|\ x\ |).`   (2 marks)

  3.    `f(x) - x.`   (2 marks)

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  1.  

     

  2.  
  3.  
Show Worked Solution
i.  
MARKER’S COMMENT: In part (ii), a significant number of students graphed  `y=|f(x)|`.
ii.

 

iii. 

Functions, EXT1′ F1 2009 HSC 3a

The diagram shows the graph  `y = f(x).`
 


 

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

  1.  `y = 1/(f(x)) .`  (2 marks)

  2.  `y = f(x)\ f(x)`  (2 marks)

  3.  `y = f(x^2).`  (2 marks)

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i.

HSC 2009 3aii

 

ii.  

iii.  

 

 

 

 

 

 

Show Worked Solution

i.   `text(Vertical asymptotes at)\ \ x=0 and 4`

`text(Horizontal asymptote at)\ \ y=-1/3`
 

HSC 2009 3aii

 

ii.  `y=f(x)\ f(x) = [f(x)]^2`

   HSC 2009 3aiii

 

iii.   `y=f(x^2) =>text(even function)`

`text(When)\ \ x=±2,\ \ y=f(4)=0`
 

 

 

 

 

 

 

Functions, EXT1′ F1 2012 HSC 11f

Sketch the following graphs, showing the `x`- and `y`-intercepts

  1.  `y = |\ x\ |- 1`   (1 mark)
  2.  `y = x(|\ x\ | - 1)`    (2 marks)

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

Graphs, EXT2 2012 HSC 11f Answer

`text{Note for part (ii)}`

`=>y = x(|\ x\ | – 1)\ \ text(is an ODD function)`

Functions, EXT1′ F1 2013 HSC 13b

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

Sketch the following curves on separate half-page diagrams.

  1.   `y^2 = f(x).`  (2 marks)

  2.   `y = 1/(1 - f(x)).`  (3 marks)

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

i.   `y^2 = f(x)`

`text(Only exist for)\ \ f(x) >= 0`

`y^2 = 1,\ \ \ y = +- 1`
 

ii.  `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. Note this important advice.

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

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

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