B.2 Functions 621The set f(C) is called the image of C under f and the set f-^1 (D) is
called the inverse image of D under f. When we write f-^1 (D) we must be
careful not to assume that f-^1 is a function. Sometimes f-^1 is a function, but
that is a separate issue, to be discussed later.
Notice that V(f) = f-^1 (B) and R(f) = f(A).
Example B.2.10 Consider the function f : JR ____, JR given by f(x) = x^2 + 3.
Let C = [-1, 2] and D = [O, 7]. Then (see Figure B.4)
f(C) = [3, 7] and f-^1 (D) = [-2, 2].
0 bserve that
f ([-1, 2]) = f ([O, 2]) = f ([-2, 2]) = [3, 7],
and that
f-^1 ([O, 7]) = f-^1 ([3, 7]) = f-^1 ([-11, 7]) = [-2, 2].
Further, f ( {2}) = f ( { -2, 2} = {7}, while f-^1 ( {7}) = f-^1 ( { -3, 0, 7} = { -2, 2}.
D
-1 1 2 xFigure B.4Theorem B.2.11 (Functions and Sets) Suppose f : A____, B is a function.
Then
(a) VC1, C2 ~A, f(C1 u C2) = f(C1) u f(C2).
(b) VC1, C2 ~A, f(C1 n C2) ~ f(C1) n f(C2).
It is possible that f(C 1 n C2)-:/::. f(C1) n f(C2). (See B.2.12 (b) below.)(c) VC1, C2 ~A, f(C1) - f(C2) ~ f(C1 - C2).
It is possible that f(C 1 ) - f(C2)-:/::. f(C1 - C2). (See B .2.12 (c) below.)(d) VD1, D2 ~ B, f-^1 (D1 U D2) = f-^1 (D1) U f-^1 (D2).