I'm asked to prove if a transformation is linear or not. In the vector field $V=\{f(x)\colon \mathbb{R} \to\mathbb{R}\}$, so the transformation is $T\colon V \to V$ given by $T(f(x)) = (xf(x))+1$. I want to show it is not a linear transformation by a counter example. Could I use the function $f(x) = x+1$, then the transformation is $T(x+1) = x^2 + x + 1$. Then, having a transformation of $T(0)$ must equal zero, so in this I would show that $x=-1 \implies T(-1+1) = T(0) = (-1)^2 + (-1) + 1 = 1 \neq 0$, showing this is not linear. I'm thinking this is right, but am not positive if this is how you would go about proving a function transformation is not linear.