show that function g is constant and f is f(x)=mx+b if f(x)=(x-y)g(x)+f(y)?

You need to prove that the function is a constant,
considering the relation provided by the problem, such
that:

y)g(x)

You need to remember the mean value theorem such
that:

, if

Since , hence
g(x)
that:

Notice that the derivative of the function is a
constant m, hence

Hence,
checking if the function g(x) is a constant, using the mean value theorem, yields that
g constant.