3.1 Functional notation 117such as h, k, p, q for Ax, Ay, tO, Az, but very often the extra labor involved
in writing the more elaborate delta symbols is a small price to pay for the
elimination of the superfluous symbols whose meanings may be forgotten
and confused.
At the conclusion of the text of this section, the author makes some
remarks that he would have made at the beginning if he had thought that
they could have been understood. The old word "function" has been
and is and will be used in many different ways. Students who get around
will have serious difficulties unless they are so well informed and tolerant
that they can accumulate and dispense information by reading and hear-
ing and talking quite different languages. It is like being able to play
football with those who play football and to play basketball with those
who play basketball; one who knows only ping-pong is sometimes handi-
capped. This, of course, does not imply that a particular teacher is
required to stand by while many different games are played simultane-
ously in his classroom. Each individual teacher may, with the full back-
ing of the author, go as far as he likes in prescribing the rules of the game
to be played in his own classroom.
Problems 3.19
1 If
f(x) = x, g(x) = xz, h(x) = 21, O(x) = 1 _+X2'1verify the following assertions and replace the question marks by appropriate
answers.(a) f(0) = 0, f(-3) -3, f(2) _?
(b) g(O) = 0, g(-2) = 4, g(5) _?
(c) h(-1) _ -, h(0) = 1, h(2) = 4, h(7) =1.4142(d) h(5) h(-2) _ ?, h(-) =?
(e) c(T) = 3, 4(2) = ?, 0(-2) = ?, O (A) _?
(f) f(8) - f(5) = 3, f(2.1) - f(2) _?
(g) g(3) - g(2) = 5, g(2.1) - g(2) _?
(h) h(3) - h(2) = 4, h(1) - h(0) _?
(i) q(1.1) - 0(1) _ -.0475, k (O.2) - (0) _?
f(4.5) - f(4) = 1f(2.8) - f(2.7)
(7) 0.5 0.1(k)g(4.1) - g(4) = 8.1,g(2.8) - g(2.7)
0.1 0.1(l)4(2.1) - 0(2)
-.152,0(0.2) - 0(0)_?
0.1 0.2(m)g(x + 2) - g(x)
= 2x + 2,g(x + 0.5) - g(x)=
2 0.5(n)g(1 +AX) - g(l) = 2 + Ax,g(ox) - g(0)
Ax dx