22. (a) Suppose that $f$ and $g$ are such functions. From $g(f(x))=x^{3}$ we have $f\left(x_{1}\right) \neq f\left(x_{2}\right)$ whenever $x_{1} \neq x_{2}$. In particular, $f(-1), f(0)$, and $f(1)$ are three distinct numbers. However, since $f(x)^{2}=f(g(f(x)))=f\left(x^{3}\right)$, each of the numbers $f(-1), f(0), f(1)$ is equal to its square, and so must be either 0 or 1. This contradiction shows that no such $f, g$ exist. (b) The answer is yes. We begin with constructing functions $F, G:(1, \infty) \rightarrow (1, \infty)$ with the property $F(G(x))=x^{2}$ and $G(F(x))=x^{4}$ for $x > 1$. Define the functions $\varphi, \psi$ by $F\left(2^{2^{t}}\right)=2^{2^{\varphi(t)}}$ and $G\left(2^{2^{t}}\right)=2^{2^{\psi(t)}}$. These functions determine $F$ and $G$ on the entire interval $(1, \infty)$, and satisfy $\varphi(\psi(t))=t+1$ and $\psi(\varphi(t))=t+2$. It is easy to find examples of $\varphi$ and $\psi$: for example, $\varphi(t)=\frac{1}{2} t+1, \psi(t)=2 t$. Thus we also arrive at an example for $F, G$:
$$F(x)=2^{2^{\frac{1}{2} \log _{2} \log _{2} x+1}}=2^{2 \sqrt{\log _{2} x}}, \quad G(x)=2^{2^{2 \log _{2} \log _{2} x}}=2^{\log _{2}^{2} x}$$
It remains only to extend these functions to the whole of $\mathbb{R}$. This can be done as follows:
$$\widetilde{f}(x)= \begin{cases}F(x) & \text { for } x>1 \\ 1 / F(1 / x) & \text { for } 0 < x < 1 \\ x & \text { for } x \in \{0, 1\} \end{cases}$$
$$\widetilde{g}(x)= \begin{cases}G(x) & \text { for } x>1 \\ 1 / G(1 / x) & \text { for } 0 < x < 1 \\ x & \text { for } x \in \{0, 1\} \end{cases}$$
and then $\quad f(x)=\widetilde{f}(|x|), \quad g(x)=\widetilde{g}(|x|) \quad$ for $x \in \mathbb{R}$. It is directly verified that these functions have the required property.