CSCE 235

Handout 9: Definitions of Functions

February 14, 2002

Basic Definitions

Let be a function from A to B.

a. The domain of f, written dom f, is the set A.

b. The target of f is the set B.

c. The range or image of f, written rng f, is

d. The function is onto (or surjective) if its range is the target, rng f = B. For any , the equation has a solution .

e. The function is one-to-one (or injective) if and only if different elements of A have different images: If , then .

f. The function is one-to-one correspondence (or bijective) if it is both one-to-one and onto.

g. For any set A, the identity function on A is the function defined by for all .

Further Definitions

a. A function has an inverse if and only if the set obtained by reversing the ordered pairs of f is a function . If has an inverse, the function

is called the inverse of f.

b. A function has an inverse if and only if f is one-to-one and onto.

if and only if .

c. If and are functions, then the composition of g and f is the function defined by for all .

d. Functions f and g are equal if and only if they have the same domain, same target, and for every a in the common domain.

e. Functions and are inverses if and only if and ; that is, if and only if

and for all and all .

Based on (Goodaire and Parmenter 2002).