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Function box

An independant variable x is passed through the function f. The value of the function is signified by f(x).

Utilization of Functions
A function is a relationship between input(s) and output(s). It can be represented as a formula or equation; for example, a formula or equation can be written as follows:

a = b 3 • c

This is valid but there exists a notation specific to functions; they can be written in the format of a pair of parentheses containing the independant variables separated by commas after the dependant variable as such:

a(b,c) = b • 3 • c

Here the relationship between the input and output is that the output is three times the input. The function [a] takes two parameters [b and c] and performs the specified operations. The final value after the evaluation is the value of a(b,c). Therefore, a(6,4) = 6 • 3 • 4 = 72 or simply, a(6,4) = 72. This statement can be represented in a natural language like English as well:

The function a of six and four is 72.

Functions can also be nested within each other's parameters, e.g:

a(i) = i + (i + 1)

f(x) = x + (x + 1)

Therefore:

f(a(1)) = f(1 + (1 + 1)) = (1 + (1 + 1)) + ((1 + (1 + 1)) + 1) = 3 + (3 + 1) = 3 + 4 = 7

This is called function composition and there is a special notation for performing it; in terms of the above case, that is:

f ∘ a(1) 

It should be noted that the common or more expected and accepted notation for an example of a function is f(x) or f of x, as f acts as an abbreviation of the word function and is the generally accepted symbol for an unknown or arbitrary independant variable.

Graphs of FunctionsEdit