Function

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Definition: Function
If [math]A [/math] and [math]B [/math] are sets, then a function from [math]A [/math] to [math]B [/math] (written [math]f : A \href{/cs2800/wiki/index.php/%E2%86%92}{→} B [/math]) is an unambiguous rule giving, for every input [math]x \href{/cs2800/wiki/index.php/%E2%88%88}{∈} A [/math], an output [math]f(x) \href{/cs2800/wiki/index.php/%E2%88%88}{∈} B [/math]. [math]A [/math] is called the domain of [math]f [/math]; [math]B [/math] is called the codomain.

When giving a function, you must indicate the domain, the codomain, and an unambiguous rule giving an output for every input.

The output of a function is unambiguous if there is only one output for any input. It is especially important to check this when the input of the function can be written in different ways. For example, [math]f : \href{/cs2800/wiki/index.php/Enumerated_set}{\{1,2\}} \href{/cs2800/wiki/index.php/%5Cto}{\to} \href{/cs2800/wiki/index.php/%E2%84%95}{ℕ} [/math] given by [math]f(1) := 2 [/math], [math]f(2) := 2 [/math], and [math]f(1+1):=4 [/math] is not unambiguous, because [math]2 = 1 + 1 [/math] so it is not clear if [math]f(2) = 2 [/math] or [math]f(2) = 4 [/math].


Specifying functions

When giving a function, you must indicate the domain, the codomain, and an unambiguous rule giving an output for every input. There are many, many ways to do this, as long as the output is clear. We gave a few examples during lecture:

direct specification

We can define the output of the function directly:

let [math]f(x) : \href{/cs2800/wiki/index.php?title=%E2%84%9D&action=edit&redlink=1}{ℝ} \href{/cs2800/wiki/index.php/%E2%86%92}{→} \href{/cs2800/wiki/index.php?title=%E2%84%9D&action=edit&redlink=1}{ℝ} [/math] be given by [math]f(x) := x^2 [/math]. The domain and codomain of [math]f [/math] are both [math]\href{/cs2800/wiki/index.php?title=%E2%84%9D&action=edit&redlink=1}{ℝ} [/math]; the image is [math]\{y ∈ \href{/cs2800/wiki/index.php?title=%E2%84%9D&action=edit&redlink=1}{ℝ} \href{/cs2800/wiki/index.php/%5Cmid}{\mid} y ≥ 0\} [/math].

drawings

We can draw a function:

Fun-a2b1c3.svg

The domain of [math]f [/math] is [math]\href{/cs2800/wiki/index.php/Enumerated_set}{\{a,b,c\}} [/math], and the codomain of [math]f [/math] is [math]\href{/cs2800/wiki/index.php/Enumerated_set}{\{1,2,3\}} [/math].

We might also write this as [math]f : \{a,b,c\} \href{/cs2800/wiki/index.php/%5Cto}{\to} \{1,2,3\} [/math] is given by [math]f(a):=2 [/math], [math]f(b):=1 [/math], and [math]f(c):=3 [/math].

tables

Another way to draw a function is with a table:

x [math]f(x) [/math]
a 1
b 1
c 1

This almost describes a function; the domain is clearly [math]\{a,b,c\} [/math] (because if there were any other domain elements, [math]f [/math] would not be a function); and the rule is clear. However, the codomain is not clear: is it [math]\href{/cs2800/wiki/index.php/Enumerated_set}{\{1\}} [/math]? Or [math]\href{/cs2800/wiki/index.php/Enumerated_set}{\{1,2,3\}} [/math]? Or [math]\href{/cs2800/wiki/index.php/%E2%84%95}{ℕ} [/math], or [math]\href{/cs2800/wiki/index.php?title=%E2%84%9D&action=edit&redlink=1}{ℝ} [/math]? When describing a function with a table, the codomain should be specified somewhere.

function description

A function need not have an algorithm for constructing the output; a description of the output is also fine. For example, let [math]dji : Days \href{/cs2800/wiki/index.php/%E2%86%92}{→} ℝ [/math] give, on input [math]d [/math], the closing Dow-Jones Industrial average on day [math]d [/math].