MATH 1090 Algebra and Trigonometry
Quiz 2 - Solutions

Note:   if you see symbols like ³ or Þ in various places, then the Symbol or WP MathA font is not installed on your computer.   The translation is:

1. Solve the inequality
   

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   Check for division by zero:
   2 - x = 0   at   x = 2.
   
   The left hand side = 0   at   x = 1
   The test intervals are   (-¥, 1], [1, 2) and (2, ¥).

   (-¥, 1] :   Try x = 0 :     Ö
   [1, 2) :   Try x = 3/2 :     ×
   (2, ¥) :   Try x = 3 :     Ö
   Therefore the solution set is
   

   or   { x | x < 1 or x > 2 }

   Alternative method:
   [The same up to]
   
   Þ     (x > 1 AND x > 2)   OR   (x < 1 AND x < 2)
   Þ     x > 2   OR   x < 1
   Þ    

2. For the function
   
   1. Find the domain of   f (x).

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      There is no division by zero in   f (x).
      To avoid the square root of a negative number in the evaluation of f, the domain of   f (x)   must satisfy   1 - x > 0 .
      Þ     1 > x
      Þ     x < 1
      Therefore the domain of   f (x)   is

      (-¥, 1]

      or   { x | x < 1 } .

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   2. Find   f ^-1(x).

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      Þ     y² = +(1 - x)
      Þ     x = 1 - y²
      Therefore

      f -1(x)   =   1 - x2

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   3. What is the domain of   f ^-1(x)?

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      The range of   f (x)   is   { y | y < 0 }
      The range of   f (x)   is also the domain of   f ^-1(x), which is therefore
      

      (-¥, 0]

      or   { x | x < 0 } .

      [Note that if the domain of the inverse function were extended to the full default domain of   g(x)   =   1 - x², then it would fail the horizontal line test.   The right half of   g(x)   is the inverse of the different function   ]

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   4. Sketch the graphs of   f (x)   and   f ^-1(x)   on the same diagram.

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      It is easier to sketch the inverse function first, as it is the left half of a parabola,
      opening downwards, with y-intercept = 1 and x-intercepts = ±1.
      The graph of   f (x)   is then its reflection in the line   y = x.
      

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[Index of Solutions]