Functions

[Pages:30]Functions

Definition

A function f from A to B is a relation from A to B such that:

(i) Dom(f) = A, and (ii) If (x,y) f and (x,z) f then y = z.

If A = B, we say that f is a function on A.

In terms of ordered pairs, (i) and (ii) say that every element of A appears as a first coordinate in one and only one ordered pair.

If f is a function and (x,y) f then we will write y = f(x).

Since f is the name of a relation we could write x f y, but absolutely no one ever writes that.

More Terminology

Functions are also called mappings.

We usually write f: A B read as "f maps A to B"

The set B is called the codomain of f.

Recall that the range of f, Rng(f) is the set of second coordinates of the relation f. So, Rng(f) B, but they need not be equal.

Example: Consider f : {1,2,3} {a,b,c,d,e} given by f = {(1,a), (2,b), (3,c)}

f is a function, Rng(f) = {a,b,c}, Codomain of f = {a,b,c,d,e}, Dom (f) = {1,2,3}, f(1) = a, f(2) = b and f(3) = c.

More Examples

Consider: f : {1,2,3,4} {a,b,c,d,e} given by f = {(1,a), (2,b), (3,c)}

This is not a function since Dom(f) A.

Consider: f : {1,2,3} {a,b,c,d,e} given by f = {(1,a), (2,b), (3,c), (1,e)}

This is not a function since 1 appears as a first coordinate in more than one ordered pair.

More Terminology

If f: A B is a function and f(x) = y we say that: y is the value of f at x; y is the image of x under f; x is the pre-image of y under f; x is an argument of f.

Notice that f(x) is the image of x under f, it is NOT the name of the function, f is the name of the function. It is an abuse of the language to call f(x) a function, although many people do.

Functions

Notice that each argument of a function has exactly one image, but an element of the codomain may have several pre-images or none at all.

Given a function f: A B we may write f = {(x, f(x)) | x A }.

This underscores the notion that a function is a set ... however, we don't often think of functions that way.

Typically, we specify a function by giving a "rule" for computing the image of an arbitrary argument such as:

f(x) = 2x -3 where x .

Note that giving the "rule" by itself does not define a function, you must also specify the domain ( x in the above example).

Real-Valued Functions

A real-valued function is a function whose codomain is .

A real-valued function of a real variable is a function whose codomain is R and whose domain is a subset of .

Real-valued functions of a real variable are so common in some branches of mathematics that we have a special convention concerning them:

If the domain of a function is not specified, it is assumed to be a real-valued function of a real variable whose domain is the largest set of real numbers for which the "rule" produces a real value.

Real-Valued Functions

Examples: f(x) = 5x2 -2x + ?

by convention is a function f : .

f

x=

x2-2 x3 x1 x-5

by convention is a function f : -{-1, 5} .

f x= x2-4

by convention is a function f : -{-2,2} .

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