How can I prove that a function F isn't a pseudo random function? Let $F$ be a length-preserving pseudorandom function. For the following constructions of a keyed function $F' : \{0, 1\}^n \times \{0, 1\}^{n−1} \to \{0, 1\}^{2n}$, state whether $F'$ is a pseudorandom function. If yes, prove it; if not, show an attack.
$$F'_k(x) = F_k(0 ∥ x) ∥ F_k(1 ∥ x) \tag{a}$$
and
$$F'_k(x) = F_k(0 ∥ x) ∥ F_k(x ∥ 1) \tag{b}$$

I don't want the solution: I want to understand how I can solve this type of exercise.
your title asks "How to prove that a function F isn't a pseudo random function", in your question case (b) is not PRF which is easy to prove. function $F'$ is PRF if for all PPT distingushers $D$, success probability of $D$ distinguishing $F'$ from random function be negligible. So for proving that function $F'$ is not PRF it suffices to construct one distinguisher (which queries to its oracle and look for non-random patterns) with non-negligible success probability.

In contrast case (a) here is PRF, which is harder to proof. for proving function $F'$ is PRF as @fkraiem mentioned suppose $F'$ is not PRF and $D'$ is distinguisher for $F'$ then construct an algorithm $A$ which breaks some assumption (Distingush $F$ from random function here breaks assumption of $F$ being PRF).