[kdtop]

I agree that this was a bit confusing. Higher up in the article, it shows that a linear function is one that doesn't change when scaled:

F(a * x) = a * F(x)

This is showing the relationship between two uses of the same function.

Then, further along, we find:

"So, what types of functions are actually linear? Plain-old scaling by a constant, or functions that look like: F(x)=ax In our roof example, a=1/3"

I think in this second situation, F(x)=ax is not a relationship but rather a DEFINITION of the function F(x).

In programming terms:

function F(x: real) : real;

begin

  Result := x * (1/3);

end;