Show that $\langle f,g\rangle= f(1)g(1)$ is not an inner product

Question

So I'm supposed to show that is not an inner product where $f,g$ are in the linear space of all real polynomials.

The problem is that I can't figure out what properties of inner product fail. They look like they all pass to me.

1. $f(1)g(1) = g(1)f(1)$ // it's commutative
2. $f(1)(g(1) + z(1)) = f(1)(g(1) + f(1) + z(1))$ \ it's distributive
3. $c(f,g) = c(f(1)g(1)) = (cf(1))g(1)) = (cf,g)$ \ it's associative
4. $f(1)f(1) > 0$ if $x \neq 0$ //it's positive

Answer 1

If $f(x) = x - 1$, then $\langle f,f\rangle = f(1)^2 = 0$ even though $f\neq 0$.