5.5 *Monotonicity, Continuity, and Inverses 275(b)
1is { positive and strictly decreasing on ( -oo, 0) if n is even; }.
negative and strictly increasing on (-oo, 0) if n is odd(c) 1(JR) = { ([O, +oo) if)n is even; }· [See proof of Theorem 5.3.16.]
-oo, +oo if n is odd
(d) if n is even, then 1: [0,+oo) ___, [0,+oo) is invertible, and 1-^1 is
continuous and strictly increasing.
(e) if n is odd, then 1: JR___, JR is invertible, and 1-^1 is continuous and
strictly increasing.- (Project) Negative Integral Power Functions: For a given n EN,
the function 1(x) = x-n is continuous on JR - {O} [see Theorem 5.1.8].
State and prove properties analogous to (a)-(e) in Exercise 12 above. - (Project) The nth Root Function: Suppose n EN. Using the inverse
function 1-^1 of the function 1(x) = xn described in Exercise 12, we define
the nth root function g by
g(x) = y'x = x~ = { 1-
1
(x) if x 2: O; }·
-1-^1 (-x) if x < 0 and n is odd
Prove that:(a) this definition of y'x is consistent with that given in Theorem 5.3.16.
(b) if x < 0 and n is odd, y'x = -v'fXI (e.g., A= --Y3).
(c) if n is even, then g : [O, +oo)
0
~
0
[O, +oo); and if n is odd, then
g : JR^0 ~
0
R In both cases, g is strictly increasing (hence 1-1) and
continuous.- (Project) Rational Exponents: First, if x =f=. 0, we define x^0 = l. If
m E Z, n EN and m and n have no common prime factor, then whenever
y'x is defined, we define x ~ = ( y'x) m.
(a) Prove that if m E Z, n E N, and m and n have no common prime
factor, then whenever y'x is defined,(b)(1) x~ = yfxm and (2) x':::t = x~, Vk EN;
Thus, Vx > 0, Vr E Q, we define xr using r = !ft as above. Note that
xr > 0.
Prove that the following "laws of exponents" hold: Vr, s E Q and
whenever xr and X^8 are defined,
(1) (xyy = xryr (3) xrxs = xr+s
(2) (xr)s = Xrs (4) Xr /xs = xr-s(5) lr = 1
1
(6) x-r = -
xr
[Use the definition of xr given above; also see Exercise 1.4.13.]