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EXAMCOPY MattTest Indenting

Consider the function , where

Part of the graph of is shown below.
 

  1. State the range of .   (1 mark)

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    1. abc
    2. def
  5. slkflskdfj
    1. slkdfjlsdkfj
    2. sdlkjfsdlkfj
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    2. sdkjflsdkfj
  7. See the items below
    1. first
    2. second
    3. third
    4. fourth
    5. fifth
    1. Find .   (2 marks)

    2. State the maximal domain over which is strictly increasing.   (1 mark)
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    2. sdlfkj
    1. sdfsdlkf
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  13. Show that .   (1 mark)

  14. Find the domain and the rule of , the inverse of .   (3 marks)
  15. Let be the function , where and .
  1. The inverse function of is defined by .
  2. The area of the regions bound by the functions and can be expressed as a function, .
  3. The graph below shows the relevant area shaded.
     

  1. You are not required to find or define .
  1. Determine the range of values of such that .   (1 mark)

  2. Explain why the domain of does not include all values of .   (1 mark)

Show Answers Only
a.    
b.i
b.ii
c.
d.
e.i 
e.ii No bounded area for
Show Worked Solution

a.   is the range.

b.i   
 
   
     
   
     

 
b.ii 

c.    
   

 
d.   To find the inverse swap and in

 
 
 
 
 
 
 

 
    Domain:
  

e.i   The vertical dilation factor of  is 

For ,

   [Using CAS]


♦♦♦♦ Mean mark (e.i) 10%.
MARKER’S COMMENT: Incorrect responses included and .

e.ii  When for  (or for  ) there is no bounded area.

  There will be no bounded area for .


♦♦♦♦ Mean mark (e.ii) 10%.

EXAMCOPY Functions, MET2 2022 VCAA 4

Consider the function , where

Part of the graph of is shown below.
 

  1. State the range of .   (1 mark)

  2.  i. Find .   (2 marks)

  3. ii. State the maximal domain over which is strictly increasing.   (1 mark)

  4. Show that .   (1 mark)

  5. Find the domain and the rule of , the inverse of .   (3 marks)

  6. Let be the function , where and .
  7. The inverse function of is defined by .
  8. The area of the regions bound by the functions and can be expressed as a function, .
  9. The graph below shows the relevant area shaded.
     

  1. You are not required to find or define .
  1. Determine the range of values of such that .   (1 mark)

  2. Explain why the domain of does not include all values of .   (1 mark)

Show Answers Only
a.    
b.i
b.ii
c.
d.
e.i 
e.ii No bounded area for
Show Worked Solution

a.   is the range.

b.i   
 
   
     
   
     

 
b.ii 

c.    
   

 
d.   To find the inverse swap and in

 
 
 
 
 
 
 

 
    Domain:
  

e.i   The vertical dilation factor of  is 

For ,

   [Using CAS]


♦♦♦♦ Mean mark (e.i) 10%.
MARKER’S COMMENT: Incorrect responses included and .

e.ii  When for  (or for  ) there is no bounded area.

  There will be no bounded area for .


♦♦♦♦ Mean mark (e.ii) 10%.

Functions, MET2 2022 VCAA 4

Consider the function , where

Part of the graph of is shown below.
 

  1. State the range of .   (1 mark)

  2.  i. Find .   (2 marks)

  3. ii. State the maximal domain over which is strictly increasing.   (1 mark)

  4. Show that .   (1 mark)

  5. Find the domain and the rule of , the inverse of .   (3 marks)

  6. Let be the function , where and .
  7. The inverse function of is defined by .
  8. The area of the regions bound by the functions and can be expressed as a function, .
  9. The graph below shows the relevant area shaded.
     

  1. You are not required to find or define .
  2.  i. Determine the range of values of such that .   (1 mark)
  3. ii. Explain why the domain of does not include all values of .   (1 mark
Show Answers Only
a.    
b.i
b.ii
c.
d.
e.i 
e.ii No bounded area for
Show Worked Solution

a.   is the range.

b.i   
 
   
     
   
     

 
b.ii 

c.  

 
 

 
d.   To find the inverse swap and in

 
 
 
 
 
 
 

 
    Domain:
  

e.i   The vertical dilation factor of  is 

For ,

   [Using CAS]


♦♦♦♦ Mean mark (e.i) 10%.
MARKER’S COMMENT: Incorrect responses included and .

e.ii  When for  (or for  ) there is no bounded area.

  There will be no bounded area for .


♦♦♦♦ Mean mark (e.ii) 10%.

Functions, MET1 2022 VCAA 5b

Find the maximal domain of , where .   (3 marks)

Show Answers Only

∪

Show Worked Solution

exists where

From the graph and

∪


♦ Mean mark (b) 53%
MARKER’S COMMENT: Students often used incorrect notation. Drawing the graph of the parabola assisted with recognition of correct domain.

Algebra, MET1 2019 VCAA 8

The function    is a polynomial function of degree 4. Part of the graph of    is shown below.

The graph of    touches the -axis at the origin.
 


 

  1. Find the rule of  .   (1 mark) 

Let    be a function with the same rule as  .

Let  , where    is the maximal domain of  .

  1. State  .   (1 mark)

  2. State the range of  .   (2 marks)

Show Answers Only
Show Worked Solution
a.   
   

 

 

b.   
   

 

c.   
   
   

  


 

Calculus, MET2 2010 VCAA 1

  1. Part of the graph of the function    is shown on the axes below

     

         

    1. Find the rule and domain of  , the inverse function of  .   (3 marks)

    2. On the set of axes above sketch the graph of  . Label the axes intercepts with their exact values.   (3 marks)

    3. Find the values of , correct to three decimal places, for which  .   (2 marks)

    4. Calculate the area enclosed by the graphs of    and  . Give your answer correct to two decimal places.   (2 marks)

  2. The diagram below shows part of the graph of the function with rule
  3.         , where , and are real constants.
     

    • The graph has a vertical asymptote with equation  .
    • The graph has a y-axis intercept at 1.
    • The point on the graph has coordinates  , where is another real constant.
       

      VCAA 2010 1b

    1. State the value of .   (1 mark)

    2. Find the value of .   (1 mark)

    3. Show that  .   (2 marks)

    4. Show that the gradient of the tangent to the graph of at the point is  .   (1 mark)

    5. If the point    lies on the tangent referred to in part b.iv., find the exact value of .   (2 marks)

Show Answers Only
  1.   i.
  2.  ii. 
     
  3. iii.
  4. iv. ²
  5.   i.
  6.  ii.
  7. iii.
  8.  iv.
  9.   v.
Show Worked Solution

a.i.  

 

ii.  

 

 iii. 

 

 

  iv.  
    ²

 

b.i.  

 

  ii.  

 

iii.  

 

  iv.  
 
   
   

 

  v.  

♦♦ Mean mark 33%.
MARKER’S COMMENT: Many students worked out the equation of the tangent which was unnecessary and time consuming.