A function $f: \{ 0,1 \}^* \rightarrow \{ 0,1 \}^*$ is called polynomial bounded if there exists a polynomial $p(n)$ such that $|f(x)| \leq p(|x|)$ for all $x \in \{ 0,1\}^*$. Here, $| x |$ means the length of $x$.
$\{0,1\}^*$ is the set of bit-strings of any size. For instance, $w_1=1101,w_2=111111101 \in \{0,1\}^*$. The length of a bit-string is defined as its number of bits. In the previous example, $|w_1|=4$ and $|w_2|=9$.
The aim is to prove that the composition of two polynomial bounded functions $f,g$ is also a polynomial bounded function.
Any hint on how the polynomial I am looking for looks like?
Well I don't know anything, but it seems to me that if $f,g$ are polynomially bounded, then there are strictly increasing polynomials $p,q$ on $[1,\infty)$ such that
$$|f(x)|\le p(|x|), |g(x)|\le q(|x|).$$
It follows that
$$|g(f(x))| \le q(|f(x)|)\le q(p(|x|)).$$
We used the fact that $q$ is increasing on $[1,\infty)$ to get the expression on the right. Since the composition of polynomials is a polynomial, $g\circ f$ is polynomially bounded.