I understand that a linear relation needs to satisfy both the property of superposition and homogeneity. Y(x)=mx+b does not satisfy both property at the same time yet any equations in this form are called a "linear function" and it is used in linear approximation. For example, sin(x), which is a non-linear function, can be approximated as sin(x0)+cos(x0)(x-x0), where b=sin(x0), m=cos(x0) and x=(x-x0), using Taylor series expansion with the operating point x0. This expression fails the linearity test, yet it is refer as a "linear" approximation of non-linear function sin(x). So if the form y=mx+b fails the superposition & homogeneity test, why do we consider it as a "linear function"? What I am missing here? Thank you so much!