- Let l(x) = 4 - x^2. Find
(a) 1J(f)
(c) l[O, 1]
(e) 1(0, 2)
(g) 1-^1 [2, 4]
(i) 1-^1 (0, 00)
(k) 1-^1 ({0}) - Let l(x) = 2x. Find
(a) 1J(f)
(c) l[O, 1]
(e) 1(0, oo)
(g) 1-^1 [1, 2]
(i) 1-^1 [-l, 1)
B.3 Algebra of Real-Valued Functions 625(b) R(f)
(d) 1-^1 [0, 1]
(f) 1-^1 (0,4)
(h) 1-^1 [-4,0]
(j) 1-^1 ( -oo, 2]
(1) 1-^1 ({-l})(b) R(f)
( d) I (-oo, 2)
(f) l[-1, ~]
(h) 1-^1 (2,8)
(j) 1-^1 (-00,0)- Redo Example B .2.12 using the function l(x) = 4 - x^2 instead of the
function given there. - Redo Example B.2.12 using the function l(x) = x^3 - 3x^2 instead of the
function given there. - Prove Theorem B.2.11 (b).
- Prove Theorem B.2. 11 (c).
- Prove Theorem B.2. 11 (d).
- Prove Theorem B.2. 11 (f).
- Prove Theorem B.2. 13 (a) (1).
- Prove Theorem B.2.13 (a) (2).
12. Prove Theorem B.2. 13 (b) (1).- Prove Theorem B.2.13 (b) (2).
B.3 Algebra of Real-Valued Functions
Definition B.3.1 Let S denote an arbitrary set. Any function I : S -+JR. is
called a real-valued function on S. We shall consider the set of all such
functions,
:F(S, JR.)= {all functions I: S -+JR.}.