Page 162 - Discrete Mathematics and Its Applications

2.3 Functions 141

DEFINITION 3 Let f 1 and f 2 be functions from A to R. Then f 1 + f 2 and f 1 f 2 are also functions from A
to R defined for all x ∈ A by

(f 1 + f 2 )(x) = f 1 (x) + f 2 (x),
(f 1 f 2 )(x) = f 1 (x)f 2 (x).

Note that the functions f 1 + f 2 and f 1 f 2 have been defined by specifying their values at x in
terms of the values of f 1 and f 2 at x.

2
2
EXAMPLE 6 Let f 1 and f 2 be functions from R to R such that f 1 (x) = x and f 2 (x) = x − x . What are
the functions f 1 + f 2 and f 1 f 2 ?
Solution: From the definition of the sum and product of functions, it follows that

2
2
(f 1 + f 2 )(x) = f 1 (x) + f 2 (x) = x + (x − x ) = x
and

2
2
4
3
(f 1 f 2 )(x) = x (x − x ) = x − x . ▲
When f is a function from A to B, the image of a subset of A can also be defined.

DEFINITION 4 Let f be a function from A to B and let S be a subset of A. The image of S under the function
f is the subset of B that consists of the images of the elements of S. We denote the image of
S by f(S),so

f(S) ={t |∃s ∈S(t = f(s))}.

We also use the shorthand {f(s) | s ∈ S} to denote this set.

Remark: The notation f(S) for the image of the set S under the function f is potentially
ambiguous. Here, f(S) denotes a set, and not the value of the function f for the set S.

EXAMPLE 7 Let A ={a, b, c, d, e} and B ={1, 2, 3, 4} with f(a) = 2, f(b) = 1, f(c) = 4, f(d) = 1, and
f(e) = 1. The image of the subset S ={b, c, d} is the set f(S) ={1, 4}. ▲

One-to-One and Onto Functions

Some functions never assign the same value to two different domain elements. These functions
are said to be one-to-one.

DEFINITION 5 A function f is said to be one-to-one, or an injunction, if and only if f(a) = f(b) implies that
a = b for all a and b in the domain of f. A function is said to be injective if it is one-to-one.