An **affine function** is a linear function plus a translation or offset (Chen, 2010; Sloughter, 2001).

Differential calculus works by approximation with affine functions. A function *f* is only differentiable at a point x_{0} if there is an affine function that approximates it near x_{0} (Chong et al., 2013).

## Formal Definition

A function f: ℝ

^{n}→ ℝ^{m}is affine if— for any x, y ∈ ℝ^{n}and for any α, Β ∈ ℝ^{m}with any α + Β = 1 — f(αx + Βy) = α f(x) + Β f(y)

The notation f: ℝ^{n} → ℝ^{m} indicates the function has a domain that’s a subset of ℝ^{m} and a range that’s a subset of ℝ^{n}. To put that another way, the function takes a vector with *m* coordinates for input and outputs a vector with *n* coordinates.

## Affine Function Examples

The familiar equation y = mx + b is usually called *linear*, but should more correctly be called *affine* (Boyd, 2007).

In fact, **every linear function is affine.**. That’s because the translation (the + b) might be the identity function (one that maps the function to itself).

## Other (Slightly Different) Definitions

Many authors define the affine function *slightly* differently, which can be a source of confusion. For example:

- “An affine function is a first degree polynomial” (Dartmouth, 2020). This is a true statement, but a polynomial of degree zero is also affine,
- De La Fuente (2000) states that “A function is affine if it is the sum of a linear function and a constant”. Also a true statement, although simplified.

In addition, an affine function is sometimes defined as a linear Form plus a number. A *linear form *has the format c_{1}x_{1} + … + c_{n}x_{n}, so an affine function would be defined as:

c_{1}x_{1} + … + c_{n}x_{n} **+ b**

Where:

*c*= a scalar or matrix coefficient,*b*= a scalar or column vector constant.

In addition, every affine function is convex *and *concave (Aliprantis & Border, 2007).

## References

Aliprantis, C. & Border, K. (2007). Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer.

Boyd, S. (2007). EE263 homework 1 solutions. Retrieved September 1, 2020 from: https://web.stanford.edu/class/archive/ee/ee263/ee263.1082/hw/hw1sol.pdf

Chen , J. (2010). Linear and Affine Functions. Retrieved May 27, 2020 from: http://www3.nccu.edu.tw/~joe/IO2010S

Christophersen, F. (2007). Optimal Control of Constrained Piecewise Affine Systems (Lecture Notes in Control and Information Sciences). Springer Berlin Heidelberg.

Chong et al., E. (2013). An Introduction to Optimization. Wiley.

Dartmouth. Best Affine Approximations. Retrieved September 1, 2020 from: https://math.dartmouth.edu/opencalc2/dcsbook/c3pdf/sec31.pdf

De La Fuente, A. (2000). Mathematical methods and models for economists. Cambridge University Press.

Hogben, L. (Ed.). (2006). Handbook of Linear Algebra. (Discrete Mathematics and Its Applications) 2nd Edition. CRC Press.

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