Primitive Recursion of two functions $g,h$


Which function is created when applying Primitive Recursion $ PR(g,h)$ to $ \begin{align}g:\mathbb{N}\to \mathbb{N}, \quad g(n)&=\mathrm{zero}_1(n)\ h:\mathbb{N}^3\to\mathbb{N}, \quad h(n,m,l)&=P_2^{(3)}(n,m,l)+\mathrm{succ}\circ P_2^{(3)}(n,m,l)+P_3^{(3)}(n,m,l) ?\end{align}$

My idea: $ f(x,0)=g(x)=zero_1(x)=0$ , $ f(x,t+1)=h(x,t,f(x,t))=t+t+1+f(x,t)=2t+1+f(x,t)$ , but what is $ f(x,t)$ and how do I derive that?

Show that if $|f(x)| \leq \phi(x) + \psi (x)$, exist $g,h$ such that $f(x) = g(x)+h(x),\, |g(x)| \leq \phi (x), \, \, \, |h(x)| \leq \psi(x)$

Given $ \phi$ and $ \psi$ two seminorms in a vector space X, and a functional $ f:X \rightarrow \mathbb{K}$ , where $ \mathbb{K} = \mathbb{R}$ or $ \mathbb{C}$ , such that $ |f(x)| \leq \phi(x) + \psi (x) \, \, \forall x \in X$ , show that there exist two linear functionals $ g,h: X \rightarrow \mathbb{K}$ such that: $ f(x) = g(x) + h(x), \, \, \, |g(x)| \leq \phi (x), \, \, \, |h(x)| \leq \psi(x) \, \, \, \forall x \in X$ .

It is obvious that if $ g,h$ exist, with the indicated bound, then $ |f(x)| \leq \phi(x) + \psi (x)$ , but how can I show that these functionals exist?