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. 

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.
