# Is calling a linear-equation a linear-function, misnomer or completely wrong?

From my college life, I remember many professors used to call a linear-equation a linear-function, however:

A standard definition of linear function (or linear map) is:

$$f(x+y)=f(x)+f(y),$$ $$f(\alpha x)=\alpha f(x).$$

Where as linear equation is defined as:

$$f(x)=mx+b.$$

So, linear-equation is NOT a linear-function, according to the definitions defined above.

Though, for $b=0$ the linear-equation becomes a linear-function, but it is not true in general.

Question: Is it misnomer to call a linear-equation a linear-function, or it is completely wrong to say that? And linear-equation must be considered strictly as an affine-mapping.

• The function $f(x)=ax+b$ is commonly called linear. In the sense of linearity as you defined it, (and it is the usual definition) then $f(x)$ is not linear if $b\ne 0$. So we have inconsistency of terminology. I think of it as essentially harmless. To call an equation of any kind a function is, however, not a good idea. Aug 21, 2014 at 3:21
• In principle, a function $f$ such that $f(x)=ax+b$ should be called an affine function. But most people will continue to call it linear. Aug 21, 2014 at 3:40
• This is the best answer math.stackexchange.com/a/217560/86137
– kaka
Aug 24, 2014 at 1:12

Neither. The meaning of some mathematical terms (normal, regular, smooth, etc) is context-dependent. E.g., smooth function means $C^\infty$ in some papers/books and $C^1$ in others.
In certain contexts, linear means "additive and commutes with scalar multiplication". In other contexts, it means "a function of the form $x\mapsto ax+b$".
One sometimes says "linear function" as verbal shorthand for "function whose value is linear in some particular variable(s)." In the case of $f(x) = ax + b$, the value $y = f(x)$ is linear with respect to the variable $x$; one might call it a linear function of $x$.
Likewise, if we define $g(x) = ax^2 + bx + c$, one would likely say that $g(x)$ was a quadratic function of $x$, or if the "of $x$" part appeared obvious enough, simply a quadratic function.