Linear functions are said to be additive: $f(x + y) = f(x) + f(y)$

But if I have this simple function $f(x)= 7x+3$, I get, for example(at $x=5$ and $8$):

$f(5)=38$ and $f(8)= 59$. The sum is $97$.

$f(5+8)= 7\cdot 13+3 = 94$.

$94\ne 97$. How come? What did I miss?

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$f$ is not linear in this case. It is affine. Linear functions from $\mathbb{R}$ to $\mathbb{R}$ look like $f(x)=cx$ where $c$ is a fixed constant. This is assuming that $f$ is continuous of course. – Daniel Montealegre Apr 2 '12 at 23:05
Linear functions in the sense of $f(x)=ax+b$ are functions whose graph is a straight line. This notion coincides with linear in the sense of additivity if and only if $b=0$. – Asaf Karagila Apr 2 '12 at 23:07
You are right, the terminologies are almost contradictory. Your function $f(x)=7x+3$ is fairly often called a linear function. But, as your calculation, it does not yield a linear transformation. – André Nicolas Apr 2 '12 at 23:12

The term linear has two distinct meanings when applied to functions.

1. A function $f(x)$ is linear in one sense if it is of the form $f(x)=ax+b$ for constants $a$ and $b$. This simply means that it is a polynomial of degree less than $2$. In graphical terms, it means that the graph is a straight line, hence the name linear.

2. A function $f(x)$ is linear in the other sense if it satisfies the condition $$f(ax+by)=af(x)+bf(y)\;.$$

The two meanings are unrelated. In particular, a linear function in the first sense is linear in the second sense if and only if $b=0$. In your example $b=3$, so while your function is linear in the first sense, it is not linear in the second sense.

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But they are not completely unrelated notions. Rather, the relationship is clarified when one studies affine spaces vs. vector spaces. – Bill Dubuque Apr 2 '12 at 23:46
@Bill: I’m perfectly well aware of that. But at the level of understanding implied by the question, they are best viewed as unrelated. – Brian M. Scott Apr 2 '12 at 23:51
So you think it is ok to make false or misleading mathematical statements so that they can be understood? – Bill Dubuque Apr 2 '12 at 23:57
FWIW, the wiki page says "...two different but related concepts." – The Chaz 2.0 Apr 3 '12 at 0:03
@Bill: I don’t consider the statement false at the level at which the question was posed. And I consider it very bad pædagogy to overwhelm students with information that is more likely to be confusing than helpful. – Brian M. Scott Apr 3 '12 at 0:07