Calculus I 
Quiz # 8
November 8, 2002
Name____________________ 
R.  Hammack 
Score ______
(1) The following function is one-to-one. Find its inverse.  [Graphics:Images/quiz8sol_gr_1.gif]

  [Graphics:Images/quiz8sol_gr_2.gif]
  
  [Graphics:Images/quiz8sol_gr_3.gif]
  
  [Graphics:Images/quiz8sol_gr_4.gif]
  
  [Graphics:Images/quiz8sol_gr_5.gif]
  
  [Graphics:Images/quiz8sol_gr_6.gif]

  [Graphics:Images/quiz8sol_gr_7.gif]
  
  [Graphics:Images/quiz8sol_gr_8.gif]
  
  [Graphics:Images/quiz8sol_gr_9.gif]

(2)   The graph of a one-to-one function f(x) is drawn. Sketch the graph of  [Graphics:Images/quiz8sol_gr_10.gif].
Reflecting across the line y = x (dashed) gives the graph of  [Graphics:Images/quiz8sol_gr_11.gif] (sketched in red).

[Graphics:Images/quiz8sol_gr_12.gif]



(3) Decide if the following functions are invertible. Explain your reasoning.

(a) [Graphics:Images/quiz8sol_gr_13.gif]

You know how to graph this function from Chapter 1.
First start with the graph of y = | x | , drawn in green.
Next graph y = | x - 2 |, drawn in orange.
Finally, graph of  y = | x - 2 | + 3, is the orange graph moved up 3 units.

[Graphics:Images/quiz8sol_gr_14.gif]


You can see that f (x) fails the horizontal line test, so it's not invertible.

(b)  [Graphics:Images/quiz8sol_gr_15.gif]
Note that  [Graphics:Images/quiz8sol_gr_16.gif]is positive for any value of x because cos(x) can get no smaller than -1, and the other terms add up to more than 10. Hence, we conclude that the graph of the function f(x) always increases, and never decreases. Therefore it must pass the horizontal line test. Thus f(x) is invertible.