TITLE: Why are linear functions linear?
QUESTION [22 upvotes]: I always thought linear functions need to satisfy
$$f(x+y)=f(x)+f(y).$$
I am a tad confused now, consider $f(x)=2x+3$. $f(1)=5$, $f(2)=7$, $f(1+2)=f(3)=9 \neq f(1)+f(2)$ which was what I thought linear functions should satisfy.
Could someone clarify?

REPLY [2 votes]: $f(x)$ = $2x$ + $3$ isn't a linear function (from and to the set of real numbers). You can easily see that $f(x+y)$ = $2x$ + $2y$ + $3$, and that $f(x)$ + $f(y)$ = $2x$ + $2y$ + $6$. Equality obviously fails.
A linear function (as a mapping from and to the set of real numbers) should be in the from $ax$, where $a$ is a constant real number.