On particular sets of primes

Question

Let $f,g:\mathbb{N}\to\mathbb{N}$ are two functions. Let $p$ be a prime such that

1) $\#\left\{p\leq x| f(p) \ \text{is also prime}\right\}=+\infty \ \text{as}\ x\to\infty$

2) $\#\left\{p\leq x| g(p) \ \text{is also prime}\right\}=+\infty \ \text{as}\ x\to\infty$

My question is: does 1) and 2) implies that

1) $\#\left\{f(p) \ \text{and} \ g(p) \ \text{are primes}\right\}=+\infty \ \text{as}\ x\to\infty$

Answer 1

No. Let the primes be enumerated as $p_i$.

Then $f(p_{2i}) = p_{2i}$, and $f(n) = 0$ otherwise, and $g(p_{2i+1}) = p_{2i+1}$ and $g(n)=0$ otherwise are counterexamples.