A Peaked Rational Family

Let

Determine, with proof, which of the following are true:

1. if $n = 2k$ ($k \in \mathbb{N}^*$), then $f(x)$ is even; 2. if $n = 2$ and $y = f(x) - k$ has two distinct zeros, then $k \in (-\infty, 0)$; 3. if $n = 1$, then for $x_1, x_2 \in (-4, 4)$ with $x_1 \neq x_2$ we always have $f(x_1) \neq f(x_2)$; 4. if $n = 1$ and the equation $kx = f(x)$ has $3$ real solutions in $(-4, 4)$, then $k \in \left(-\infty, -\dfrac{1}{4}\right)$.