706 Appendix C • Answers & Hints for Selected Exercises
- (a)~ (b) (-oo,4] (c) [3,4] (d) [-2,-v'3]u[v'3,2]
(e) (2, 4)
(i) (-2, 2)
(f) (-2,2) (g) [-vl2,V2] (h) [-v's,-2] u [2,v's]
(j) (-oo, -V2] u [vi2, oo) (k) { -2, 2} (1) {±vs}- (a)~
(f) [~, V2]
(b) (0, oo)
(g) [O, 1](c) [1,2]
(h) (1, 3)(d) (0, 4)
(i) (-oo, 0)(e) (1, oo)
(j) 0- (a) (-4,0), (-4,0) (b) 0, (-4,0) (c) (-4,0], (-4,-2]
(d) (-oo, 16), (-oo, O] u (f-i(2), 1-i(4))
(e) u-i(2), 1-i(3)]' u-i(2), 1-i(3)], (f) (-oo, ri(2)], (-oo, 1-i(2)) - Let y E l(Ci) - l(C2). Then, 3x E Ci 3 y = l(x). But x i C2 since
l(x) = y i l(C2). :. x E Ci - C2, soy E l(Ci - C2). - x E 1-i(Di - D2) {::} l(x) E Di - D 2 {::} l(x) E Di and l(x) i D2 {::}
x E 1-i(Di) and xi 1-i(D2) {::} x E 1-i(Di) - 1-i(D2). - y E 1 ( n C;..) =* 3x E n C;.. 3 l(x) = y
-.EA >-.EA
=? 3x 3 \:/>.EA, x EC;.. and l(x) = y
=;.\:/>.EA, 3x 3 x EC;.. and l(x) = y (Note: this step is not{::})
=?\:/>.EA, y E l(C;..) =* y E n f(C;..).
-.EA
- x E 1-i ( n D;..) {::} f(x) E n D;.. {::} \:/)...EA, l(x) ED;..
-.EA >-.EA
{::}\:/)...EA, x E 1-i(D;..) {::} x E n f-i(D;..).
-.EA
EXERCISE SET B.3- (J+g)(x) = x^2 +2x-1 (J-g)(x) = -x^2 +2x+3 f(x+2) = 2x+5
f(x) + 2 = 2x + 3 g(x + 2) = x^2 + 4x + 2 g(x) + 2 = x^2
3l(x)=6x+3 f(3x)=6x+l 3g(x)=3x^2 -6 g(3x)=9x^2 -2
(Jg)(x) = 2x^3 + x^2 - 4x - 2 (J /g)(x) = (2x + 1)/(x^2 - 2)
lfl(x) = l2x + ll max{f,g}(x) = { x2 - 2 if x:::; -1 or x ~ 3}
2x + 1 if - 1 :::; x :::; 3min. {f ,g }() x = {2x+lifx::;-lorx~3}
x^2 - 2 if - 1 :::; x :::; 3
(f o g)(x) = 2x^2 - 3 (go l)(x) = 4x^2 + 4x - 1- (a) D (f) = R(f) = ~; 1-1 (b) D (f) = [-1, oo); R(f) = [O, oo); 1-1
(c) D (f) = (-oo, -1] U [1, oo); R(f) = [O, oo); not 1-1
(d) D(J) = (O,oo); R(J) = ~; 1-1 (e) D(f) = ~; R(f) = (O,oo); 1-1
(f) D (f) = (-oo, -1) U (-1, oo); R(f) = (-oo, 1) U (1, oo); 1-1
(g) D (f) = ~; R(f) = [-1, 1]; not 1-1 (h) D (f) = R(f) = ~; 1-1