FUNCTIONS

Input, Process, Output. Transforming raw algebra into predictable mathematical machines and mapping engines.

01 // The Machine

A function is a strict relationship rule: every input is allowed to have exactly one output. It is a completely predictable machine. If you put in the number 2, you must get the exact same answer every single time.

Domain

The entire set of valid Inputs (

x

). The fuel for the machine.

Range

The entire set of possible Outputs (

y

f(x)

). The result.

Input

(x)

f(x)

Output

(y)

02 // The Routing Logic

The Mapping Engine

Select Machine

f(x)

Test Input (x)2

f(2)2

Domain

Inputs (x)

Range

Outputs f(x)

03 // Validation Protocol

Vertical Line Test

Status: PASS

The vertical scanner hits the graph exactly once at any given time. Every $x$ has a unique $y$ . It is a Function.

Vertical Line Test

Status: FAIL

The vertical scanner hits the graph at two points simultaneously. A single $x$ cannot yield two different $y$ values.

Function Inception

Functions can be stacked. By feeding the completed output of one machine directly into the input of a second machine, we create a Composite Function.

g(x)

)

Input

x

Execute

g(x)

Pass Result

Execute

f(x)

Machine Logic Mastered

You are ready to command continuous logic.

Next: Exp & Logs