Discrete Mathematics for Computer Science

238 CHAPTER^4 Functions

is decreasing on (-oo, 3] and increasing on [3, 00). (You can see this from the graph of the

function.) The definition of the terms increasing and decreasing uses the familiar orderings

less than and less than or equal on R.

Definition 9. Let X, Y C R, and let F : X --> Y be a function.

(a) F is increasing if for, all x1, x2 E X, Xl < X 2 implies F(x 1 ) < F(x 2 ).

(b) F is strictly increasing if, for all x1, X2 E X, xl < x2 implies F(x 1 ) < F(x 2 ).

(c) F is decreasing if, for all Xl, x2 E X, x1 < x2 implies F(x 1 ) > F(x 2 ).

(d) F is strictly decreasing if, for all xl, X2 E X, x1 < X2 implies F(x 1 ) > F(x 2 ).

Example 18. The following functions are increasing:

(a) The function F : R --* R where F(x) = x^3 is strictly increasing (see Figure 4.19).

y

0.6--

0.4

0.2

-3 -2 -1 1 2 3

.2-

(-0.4

_-0.6

Figure 4.19 F(x) x^3.

(b) The function Floor :R -N N is increasing but not strictly increasing (see Figure 4.20).

y

o points not included

2 - in the line

0o-

I I I x

-3 -2 -1 1 2 3

-- --- 1

o -- 2

0- -3

Figure 4.20 Floor

Theorem 2. Suppose X C R and F : X -- JR is a strictly increasing function. Then,

F is 1-1.