Linear Function: f(ax + by) = af(x) + bf(y) vs y = ax+b.

Question

I studied a linear function definition in our Machine Learning course:

f(ax + by) = af(x) + bf(y)

Using this definition, we can prove that y = ax + b is NOT a linear function. This is counter-intuitive to me.

For example: f(x) = 2x + 3 is NOT linear.
f(0 + 1) = f(1) = 5
but f(0)+f(1) = 3+ 5 = 8. 
Hence f(0+ 1) != f(0)+f(1)

We all know that y = ax + b is a line function (I know this is a linear function for ages), but it is not linear according to this definition. I am very confused.

In what situation we should use the above definition?

Recourse:

https://homepage.divms.uiowa.edu/~idarcy/COURSES/34/linearfn2s.pdf

Answer 1

The equation $y=ax+b$ is the composition of a linear function $y=ax$ with a translation $b$. Even if $y=ax+b$ is not linear it preserves some properties of the related linear function and it is denoted as an affine function and more precisely as an affine line in that case.

Answer 2

linear function $\neq$ linear polynomial

The former is what you have defined in your Machine learning course. The latter is functions like $ax+b$. See wikipedia for more details and context.

Linear functions are very important in the subject of linear algebra.