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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 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 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 2016 HSC 11d

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

ext2-hsc-2016-11d

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

  1.  `y = sqrt (f(x))`  (2 marks)

  2.  `y = 1/(f(x))`  (2 marks)

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i.   

ext2-hsc-2016-11d-answer2
ii.   

ext2-hsc-2016-11d-answer4
Show Worked Solution
i.    ext2-hsc-2016-11d-answer2

 

ii.    ext2-hsc-2016-11d-answer4

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 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