TITLE: Should a linear function always fix the origin?
QUESTION [8 upvotes]: I became very confused about linear functions after reading this question What is the difference between linear and affine function
In the comments it says that $F(x)=2*x+4$  is NOT a linear function , (but an affine one). All my professors gave such examples when teaching linear functions. I am really confused now.
Should a linear function always be of the form $f(x)=t*x$ , where t is a constant ?
I think this could help me understand better linear transformations. I think one of the reasons I did not understand them is because I had a slightly wrong definition of linear functions.
HOWEVER, on Wikipedia, the definition of linear functions seems to accepts functions that also have a constant added or subtracted from the first (linear?) part.
https://en.wikipedia.org/wiki/Linear_function
https://upload.wikimedia.org/wikipedia/commons/0/0e/Linear_Function_Graph.svg
So is wikipedia wrong on this one ?

REPLY [3 votes]: If $L$ is a linear function, then $L(0)=L(x-x)=L(x)-L(x)=0$. So, a linear function always fix the origin.

REPLY [2 votes]: A  function defined from $ \mathbb R \to \mathbb R$ is said to be linear if 

$\forall x, y\; \;\;f(x+y)=f(x)+f(y)$
$\forall \lambda \;\;\;f(\lambda x)=\lambda x$

so, $x\mapsto ax$ satisfies these conditions while $x\mapsto ax+b$ does not and it  is affine.