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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.
           

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

 

ii. 

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

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

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