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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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    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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    @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