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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.2012-04-02
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    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 $b=0$.2012-04-02
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    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*.2012-04-02

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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 different (though related in other complex ways). 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](http://en.wikipedia.org/wiki/Affine_space) vs. [vector spaces.](http://en.wikipedia.org/wiki/Vector_space)2012-04-02
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    @Bill: I’m perfectly well aware of that. But at the level of understanding implied by the question, they are best viewed as unrelated.2012-04-02
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    So you think it is ok to make false or misleading mathematical statements so that they can be understood?2012-04-02
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    FWIW, [the wiki page](http://en.wikipedia.org/wiki/Linear_function) says "...two different but *related* concepts."2012-04-03
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    @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.2012-04-03
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    @Brian I think it would have been better to write something like "the two meanings can be related from a more advanced standpoint" rather than to state as fact that "the two meaning are unrelated". One can severely hamper mathematical intuition by instilling false beliefs in budding mathematical minds.2012-04-03
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    @Bill: Indeed. So it’s a good thing that my answer is unlikely to do so, isn’t it?2012-04-03
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    @Brian Obviously, if I agreed with that I would not have written what I did above. I was hoping that you might rephrase the remark to be a bit less misleading.2012-04-03