So I was sitting in calculus class today and flipping through my text book and had to do a double take to make sure I wasn't going crazy:
On August 26, 2005, the runner Kenenisa Bekele of Ethiopa set a world record for the 10,000-meter race. His time, in seconds, at 2000-meter intervals are recorded in table 1.11, where t= f(d) is the number of seconds Bekele took to complete the first d meters of the race. For example, Bekele ran the first 400 meters in 629.98 seconds, so f(4000) = 629.98. The function f was useful to athletes planning to compete with Bekele.
Table 1.11: Bekele's Running Time
d(meters) ------ t = f(d) (seconds)
---------------------------------------
0.................................. 0.00
2000............................ 315.63
4000............................. 629.98
6000............................. 944.66
8000............................. 1264.63
10000........................... 1577.53
Let us now change our point of view and ask for distances rather than times. If we ask how far Bekele ran during the first 629.98 seconds of his race, the answer is clearly 4000 meters. Going backward in this way from numbers of seconds to numbers of meters gives f^-1, the inverse function of f. We write f^-1 (629.98) = 4000. Thus, f^-1( t ) is the number of meters that Bekele ran during the first t seconds of his race. See table 1.12 which contains values of f^-1.
Table 1.12: Distance run by Bekele
t(seconds) ------ d = f^-1(t) (meters)
---------------------------------------
0.00 .................................. 0.00
315.63 .............................. 2000
629.98............................... 4000
944.66............................... 6000
1264.63............................. 8000
1577.53............................. 10000
The independent variable for f is the dependent variable for f^-1, and vice versa. The domains and ranges of f and f^-1 are also interchanged. The domain of f is all distances d such that 0< and = d< and = 10000, which is the range of f^-1. The range of f is all times t, such that 0< and = t< and = 1577.53, which is the domain of f^-1.
I was totally not expecting to see him in my book, pretty cool though. It will be 6 years to the day tomorrow that he set the record that is mentioned in my book, just 2 days before he will look to defend his WC title!