Determine if its linear the transformation
$f:\Re_{n}[x] \rightarrow \Re$
such as
$f(p(x))=p(x)+1$
for any $p(x) \in \Re_{n}$[x]
The solution says it's not linear but I worked it out like this:
Condition 1. $T(u+v)=T(u)+T(v)$
$f(p(x))+p'(x))=((p(x)+1)+(p'(x)+1))=f(p(x))+f(p'(x))$
Condition 2. $\alpha T(u)=T(\alpha u)$
$\alpha f(p(x))= \alpha (p(x)+1) = \alpha p(x)+\alpha = f(\alpha p(x))$
What am I doing wrong?
It should be $$f(p(x)+p'(x)) = p(x)+p'(x)+1 \ne p(x)+p'(x)+2 = f(p(x)) + f(p'(x))$$
Anyway, it is faster to note that $f(0) = 1 \ne 0$, so $f$ cannot be linear.