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).

Examples from the video

• 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.

How to check whether a relation is a function

1. Identify the input variable (x) and the output variable (y).
2. 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.
3. (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.

Takeaway

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).

Speakers / sources featured

• 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)