25(1 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8real numbers, tbese operations are so general as to be applicable to any
mappings. The operations are inverse, restriction, and composition of func-
tions and mappings.
The inverse of a function f is simply the inverse f - ' = ((y, x) 1 y =
f(x)} off regarded as a relation, as defined in Article 7.1, Definition 5.
As examples, iff = ((3, 5), (5,8), (7, 1 I), (9, 14), (1 1, 17)), then f -' = {(5, 3),
(8,5),(11,7),(14,9),(17,11)}. If g=((x,4~-7)Ix~R}, then g-'=
((4x - 7, x)lx E RJ, which, as you should verify, is the same as {(y, y/4 -
714) 1 y E R), or ((x, x/4 - a 1 x E R}. The properties dom (f - ') = rng f,
rng (f - ') = dom f, and (f - ')- ' = f, listed in Theorem 3, Article 7.1, are,
of course, true in the special case of a relation that happens to be a function.
Iff maps R into R, then the graph off -' is simply the "mirror image"
of the graph off across the line y = x (see Figure 8.2). Familiar examples
from your calculus experience are the logarithm and exponential functions
to various bases, the nth power and nth root functions, and the trigono-
metric and hyperbolic functions together with their inverses. The function
f(x) = l/x provides us 'with an unusual situation; this function is its own
inverse as shown in Figure 8.3.
The key question concerning inverse functions is: "Under what circum-
stance(~) is the inverse of a function itself a function?" In order for f -' to
be a function, it must be the case that no two distinct ordered pairs of
elements in f - ' have the same first element. This means that, back in
Figure 8.2 There is a spec@ relationship between the graph of
a function and the graph of its inverse. The graphs off and f -
are symmetric with respect to the line y = x.
y =f-l(x)