Suppose that the functions $u$ and $w$ are defined as follows.
$$\begin{array}{l}u(x)=x^2+1\\ w(x)=\sqrt{x+6}\end{array}$$
Find the following.
$$\begin{array}{l}(u\circ w)(3)=\\ (w\circ u)(3)=\end{array}$$

Step 1 ：Define the functions $u$ and $w$ as $u(x)=x^2+1$ and $w(x)=\sqrt{x+6}$ respectively.

Step 2 ：The notation $(u\circ w)(x)$ means $u(w(x))$, and $(w\circ u)(x)$ means $w(u(x))$.

Step 3 ：To find $(u\circ w)(3)$, we first need to find $w(3)$ and then substitute that into $u(x)$.

Step 4 ：To find $(w\circ u)(3)$, we first need to find $u(3)$ and then substitute that into $w(x)$.

Step 5 ：Calculate $w(3)$: $w(3)=\sqrt{3+6}=\sqrt{9}=3$

Step 6 ：Substitute $w(3)$ into $u(x)$: $u(w(3))=u(3)=(3{)}^2+1=10$

Step 7 ：So, $(u\circ w)(3)=\boxed{{\displaystyle 10}}$

Step 8 ：Calculate $u(3)$: $u(3)=(3{)}^2+1=10$

Step 9 ：Substitute $u(3)$ into $w(x)$: $w(u(3))=w(10)=\sqrt{10+6}=\sqrt{16}=4$

Step 10 ：So, $(w\circ u)(3)=\boxed{{\displaystyle 4}}$