Problem One: 2000-2001
 Calculus I (AB): Functions Defined by Graphs

    Given the functions f(x) and g(x) as defined by the two graphs below:

(a)    Which of the two functions has an inverse? Why?
(b)    Sketch the inverse of the function you named in part (a).
(c)    Evaluate g(f(0)).
(d)    Draw the graph of |f(x)|.
(e)    Draw the graph of g(|x|).
Difficulty Rating: 
Solution:
    (a)    g(x) has an inverse because it passes the horizontal line test, and is therefore one-to-one.
    (b)    To graph the inverse function, switch the x and y coordinates of all the major points on the graph of g(x) to get:
    (c)    According to the graph of f, f(0) = 2, since the graph contains the point (0,2). For the same reason, g(2) = 1. Therefore, g(f(0)) = 1.
    (d)    This graph is the same as f, but cannot have any negative y values. Graphically, this means that the negative portion of the graph is "flipped" above the x-axis.
    (e)    When the absolute value signs affect the input like this, the graphical effect is the elimination of the graph for x < 0, replacing it with a y-symmetric copy of x > 0.
Calculus II (BC): Infinite limits
    Design two functions, f(x) and g(x), which satisfy both of the following conditions:
* 
* 
Difficulty Rating: 
Solution:
The simplest solution to this dilemma are two functions of equal degree. Therefore, the limit of their quotient (as x approaches infinity) will simply be the ratio of the leading coefficients. One (of many possible) solutions is:
Both of these functions approach infinity as x grows infinitely large, and the limit of the quotient, as x approaches infinity, is 3, since the ratio of the leading coefficients is 3/1.

Home Problem Fun Calculus Stuff Kelley's Books Superbowl of Calculus