Greenwood press

44
INVERSE
(
MULTIPLICATIVE
)

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Gear ratios


Investigating direct and inverse variation with the telescope
elescope.html>
Murphy’s law


Weight and distance on a lever



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INVERSE FUNCTION
An inverse is a process, procedure, or operation that is reversed. For exam-
ple, the inverse of walking up the stairs is walking down the stairs. The inverse
of putting on your socks and then your shoes in the morning is taking them off
at night. When you are given driving directions to a friend’s house, you have to
use the inverse of the original directions to find your way home, because all of
the directions will need to be reversed, where left turns will become right turns,
and vice versa. 
Two functions, 
f (x) and g(x), are inverses if their composites are equal to the
independent variable. Symbolically, this is written 
f (g(x)) = x or g(f (x)) = x.
Also, the coordinates on inverse functions are reversed. If 
f (x) and g(x) are
inverses, and 
f (x) contains the point (4,7), then g(x) contains the point (7,4). So
one way to model an inverse of a function is to reverse the coordinates. For exam-
ple, the exchange rate when traveling from the United States to Australia might be
represented by the function 
a = 1.90u, where u is the number of U.S. dollars, and
a is the number of Australian dollars. This means that 1.90 Australian dollars is
equivalent to 1 U.S. dollar. In this equation, the coordinates are represented by the
ordered pair 
(u, a). The inverse of this relationship would be to describe the
exchange rate when traveling from Australia to the United States. Therefore the
coordinates would be reversed, or 
(a, u). In order to represent this equation as a
function that indicates the exchange rate, the equation 
a = 1.90u needs to be
rewritten as a function in terms of 
a. This can be done by dividing both sides of
the equation by 1.90 to obtain 
1a
1.90
=
1.90u
1.90
, which simplifies to approximately
0.53a = u. This means that the exchange rate on the return to the United States is
about $0.53 (U.S.) for every Australian dollar. You can verify that these two func-