Summary of "함수, 일차함수 (개념01) | 중2수학"

Definitions and core ideas

Function (intuitive): a rule or “machine” that assigns to each input x exactly one output y. Written as f(x) = y.
Formal requirement: for every allowed input x there must be one and only one corresponding y.
Linear function: a function in which y is a linear expression in x (degree 1). Typical form: y = ax + b (a and b are constants).

Doubling machine: f(x) = 2x. Every input x produces a single, well-defined output 2x → this is a function.
Squaring machine: f(x) = x^2. Every input x produces a single output x^2 → this is a function, but not a linear function (degree 2).
Capsule vending machine: inserting the same coin (same input) returns a randomly chosen capsule (different possible outputs for the same input). This is NOT a function because the output is not uniquely determined by the input.

Identify the input variable (x) and the output variable (y).
For each possible x, determine whether the rule or mechanism gives exactly one y:
- If each x has one and only one y → it is a function.
- If some x can produce multiple different y values → it is not a function.
(To check linearity) If it is a function, examine the form of y in terms of x:
- If y is degree 1 (an expression ax + b) → it is a linear function.
- If y contains higher powers (x^2, x^3, etc.) or other nonlinear operations → it is not linear.

A function is a unique mapping from input x to output y. A linear function is a special function where y depends on x linearly (degree 1).

Narrator / teacher (unnamed) — main speaker explaining concepts
Background music (audio track)
Example “sources” used in the explanation: doubling machine (f(x) = 2x), squaring machine (f(x) = x^2), capsule vending machine (random output example)