B.3 Algebra of Real-Valued Functions 629(12) Let f , g, h E F(S, JR). Then, \:/x ES,
[f(gh)](x) = f(x) · (gh)(x) by definition of f(gh)
= f(x) · [g(x) · h(x)] by definition of gh
= [f(x) · g(x)] · h(x) by axiom (M2) of JR
= (f g)(x) · h(x) by definition off g
= [(fg)h](x) by definition of (fg)h.Thus, by Definition B.2.2, f(gh) = (fg)h.
(13) See Exercise 5.
( 14) Exercise 11.
(15) Exercise 12. •Students who have had a course in linear algebra will observe that Proper-
ties (1)- (10) say that F(S, JR), together with the addition and multiplication
by "scalars" in Definition B.3.1, is a vector space. Students who have also had
a course in abstract algebra will observe that properties (1)-(5) and (11)-(14)
say that F(S, JR), together with the addition and multiplication in Definition
B.3.1, is a commutative ring. They may be interested in proving that F(S,
JR), together with the addition and multiplication in Definition B.3.1, is not an
integral domain.
Properties (1)-(15) taken together, say that F(S, JR), together with the
addition, multiplication by "scalars,'' and multiplication in Definition B .3.1, is
a commutative algebra. The theory of "algebras" is important in advanced
analysis, but not in this course.COMPOSITE FUNCTIONS AND INVERSES
Definition B.3.4 If f: A--+ Band g: B--+ C, then the composite function
g o f is defined by the rule
\:/x ES, (go f)(x) = g(f(x)).
The following schematic diagram may be helpful in giving an intuitive
understanding of g o f:gofFigure B.7