Let $U = \{(s,0) | s\in \Bbb R\}$, $V = \{(s,0)| s <0\} \cup \{(s,1)|s>0\}$ and $$\phi: U \to \Bbb R, \psi: V \to \Bbb R, \gamma: V \to \Bbb R$$ be respectively given by $$\phi(s,0) = s,\psi(s,0) = s,\psi(s,1) = s , \text{and} \ \gamma(s,0) = s^3,\gamma(s,1) = s^3.$$ To show that $\{(U,\phi),(V,\psi),(V,\gamma)\}$ defines a smooth manifold structure on $U \cup V$.
How to proceed with the problem? Thank You.
We need to show the compatibility, which means that for all pairs of charts $(A,f)$ and $(B,g)$ we have $f\circ g^{-1}:g(A\cap B)\rightarrow f(A\cap B)$ is smooth, in addition to the requirement that charts must cover our manifold.
Let's show that $(V,\psi)$ and $(V,\gamma)$ are compatible. Your map $\psi \circ \gamma ^{-1}=s^{\frac13}:\mathbb{R}-\{0\}\rightarrow \mathbb{R}-\{0\}$ is clearly smooth. Similarly, $(V,\gamma)$ and $(V,\psi)$, because $s^3$ is smooth.
Can you proceed for the remaining maps, namely $\phi\circ\gamma^{-1}$, $\phi\circ\psi^{-1}$, and their inverses $\gamma\circ\phi^{-1}$, $\psi\circ\phi^{-1}$ ?
Btw, be careful about the domains of your maps. For some cases we are lucky that $V$ doesn't contain $0$.