Example. Let $y=f(u)=\lg u, u \in R^{+}$, $u=\varphi(x)=\sin x, x \in R$. Try to discuss the domain of the composite function $y=f[\varphi(x)]$.
To make the function $y=f(u)=\lg u$ defined, it is necessary to have $u>0$, i.e., $\sin x>0$, which means $x \in(2 k \pi, (2 k+1) \pi), k \in \mathbb{Z}$.

Therefore, the domain of the composite function $y=f[\varphi(x)]=\lg \sin x$ is: $\{x \mid 2 k \pi < x < (2 k+1) \pi, k \in \mathbb{Z}\}$.
The domain of $u=\varphi(x) \times \sin x$ is: $x \in \mathbb{R}$.
From this, we can see: the domain of the composite function $y=f[\varphi(x)]$ and the domain of $\varphi(x)$ may not be the same.

For example, the domain of $y=f[\varphi(x)]=e^{x^{2}}$ and $\varphi(x)=x^{2}$ are the same, both being $x \in \mathbb{R}$.

It can be seen that the domain of the composite function $y=f[\varphi(x)]$ (let it be set $A$) and the domain of $u=\varphi(x)$ (let it be set $B$) may be the same or different. However, the domain of the composite function is always a subset of the domain of $\varphi(x)$, i.e., $A \subseteq B$.