1549901369-Elements_of_Real_Analysis__Denlinger_

B.3 Algebra of Real-Valued Functions 627

(g) Minimum: V f, g E F(S, JR), we define the function min{!, g}

by specifying that Vx E S ,

min{f,g}(x) = min{f(x),g(x)}.

"--v--'

in F(S,JR) in JR

Example B.3.2 Consider the functions f,g E F(JR,JR) given by f(x) = 3x+2

1

and g(x) =--.Then Vx E JR,

x-1

1

(a) (! + g)(x) = f(x) + g(x) = 3x + 2 + --.

x-1

(b) 2f(x) = 2(3x + 2) = 6x + 4.

(c) (fg)(x) = f(x)g(x) = (3x + 2) ( - 1 ) = --. 3x +^2

x-1 x-1

(d) lfl(x) = lf(x)I = l3x + 21.

(e) max{f,g}(x) = max{3x+2,-

1
-}; for example, max{f,g}(O) = 2
x-1
1
and max{f,g}(-1) = -
3

.

(f) min{f, g}(x) =min { 3x + 2, x ~

1

} ; for example, min{f,g}(O) = -~

and min{f,g}(-1) = -1. D

Theorem B.3.3 (Algebra of Functions) Let S denote an arbitrary nonempty
set. Then F(S, JR), together with the operations (a)-(c) specified in Definition
B .3.1 above, satisfies the following properties:

(1) Vf,g E F(S, JR), f + g E F(S, JR);

(2) Vf,g,hEF(S,JR),f+(g+h)=(f+g)+h;

(3) Vf,gEF(S, JR), f+g=g+f;

(4) 30EF(S,JR)3VfEF(S,JR),f+0=0+f=f;

(5) VJ E F(S, JR), 3 -f E F(S, JR) 3 f + (-!) = O;

(6) VJ E F(S, JR), V r E JR, rf E F(S, JR);

(7) Vf,g E F(S, JR), V r E JR, r(f + g) = rf +rg;

(8) VJ E F(S, JR), V r , s E JR, (r + s) (!) = rf +sf;

(9) VJ E F(S, JR), V r,s E JR, r(sf) = (rs)f = s(rf);

(10) VJ E F(S, JR), lf = f;