Affine functions and degree-one functions

Mathematical statement

For m,b∈R, the function f:R→R defined by f(x)=mx+b is affine. If m≠0 its polynomial degree is one; if m=0 it is constant. An R-linear map in linear algebra additionally requires b=0, and must be distinguished from the elementary textbook term linear function.

Representative formula (exact TeX)

f:R\to R

Assumptions

• The domain and codomain are both R, and m and b are real constants.
• A two-point slope is defined only for x_1≠x_2. Comparison at a single input concerns only the outputs at that input.

Symbol scope

• f — real affine function
• R — real domain and codomain

Intuition, definition, and misconceptions

m is the change in output per unit change in input, and b=f(0). Two graph points with different first coordinates give m=(y_2−y_1)/(x_2−x_1); a vertical line does not allow division by zero.

For m≠0 there is a unique zero, −b/m. If m=0 and b=0, every real number is a zero; if m=0 and b≠0, there are none. Constant functions belong to the affine family but are not degree-one functions.

Comparing f(x_0) and g(x_0) establishes only a fact at that input. Two affine functions on all of R agree exactly when both slopes and both intercepts agree. A single-input exercise in this pulse cannot be promoted to global equality.

Common errors include calling every mx+b degree one, or treating an affine function with a nonzero intercept as a linear map in linear algebra.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Function mapping — Specify the inputs, outputs, and identity of the function on its whole domain.
• Linear equation in one variable — When m≠0, a zero is obtained by solving a linear equation.

Verification boundary

Integer-coefficient evaluations and rational two-point slopes can be recomputed exactly; the general affine structure and linear-map bridge await formalization.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind affine functions over R, the m≠0 degree-one condition, constant-function zero sets, and the homogeneity boundary for linear maps.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.