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

你的問法有問題﹐你應該想問operator
一個operator L 可以將一個函數f映射去另一個函數g
若L(af+bg)=aL(f)+bL(g)
則L是linear
中一至中七的例子有d/dx和矩陣A
inverse opreator of a linear opreator must be linear
SINCE
L[L^-1(f)+L^-1(g)]
=L[L^-1(f)]+L[L^-1(g)]
=f+g
=L[L^-1(f+g)]
So L^-1(f+g)=L^-1(f)+L^-1(g)

you can try it...
let f(x)=x^2+2x+1
f(ax+by)
=(ax+by)^2+2(ax+by)+1
=(ax)^2 +2abxy+(by)^2+2ax+2by+1

af(x)+b f(y)
=a(x^2+2x+1)+b(y^2+2y+1)
=ax^2+2ax+a+by^2+2y+1
not equal to f(ax+by)

so f(ax+by)=af(x)+b f(y) this statment is not ture