XVI Functions

A function in math is a rule (made up of a set of operations on a number) that describes how a given number may be transformed.

Function Diagram

The expression 3n+13n + 1 can be described in the format:

n×3+13n+1n \rarr \boxed{\times 3} \rarr \boxed{+1} \rarr 3n+1

Which can be written more explicitly:

n×33n+13n+1n \rarr \boxed{\times 3} \rarr 3n \rarr \boxed{+1} \rarr 3n+1

Mapping/Arrow Diagram

A rule may be described by a mapping of inputs to its outputs:

152739411513615}\begin{rcases} 1 \rarr 5 \ 2 \rarr 7 \ 3 \rarr 9 \ 4 \rarr 11 \ 5 \rarr 13 \ 6 \rarr 15 \end{rcases} x2x+3x \rarr 2x+3


The set of valid input numbers of a function is called its domain while the set of possible output numbers is called its codomain.

Domain and codomain of a function are not always explicit.


Functions may be written in a number of ways:

Functional Notation

Ambiguous notation that implicitly uses the set of real number R\reals for its domain and codomain.

Given the function definition f(x)=x+1f(x) = x + 1

f(x)f(x) becomes shorthand for x+1x + 1

f(x)f(x) is the image of x of ff

f(5)=6f(5) = 6

This notation is ambiguous because f(x)f(x) could refer to f(x)=x+1f(x) = x + 1 or an image of ff for a certain number if one has previously been declard for xx.

Arrow Notation

Also implies a domain and codomain of R\reals. This notation is used to define rules inline without naming them.

xx2+4x \rarr x^2 + 4

Index Notation

Is often used instead of functional notation. It uses a subscript instead of parantheses to name its input.


This notation is typically used for sequences of natural numbers.

Composite Functions

The output of one function f(x)f(x) can be directly channeled into the input of another function g(x)g(x) by combining their letters.

f(x)=x+2f(x) = x + 2, g(x)=x2g(x) = x^2

gf(x)=(x+2)2gf(x) = (x+2)^2 (the innermost function is applied first)

Inverse Function

An inverse function runs a function's opposite operations in reverse order.

n×3+13n+1n \rarr \boxed{\times 3} \rarr \boxed{+1} \rarr 3n+1

n÷31n13n \larr \boxed{\div 3} \larr \boxed{-1} \larr \frac {n-1}{3}

The inverse to f(5)=11f(5) = 11 is f1(11)=5f^{-1}(11) = 5

f(x)=x+6f(x) = x + 6

f1=x6f^{-1} = x - 6

Finding the input of a given output

If a function's rule and an output are known, that output can be traced through the function's inverse rule to find its original input (domain).