Identify the outer function.

The given function $f(x)$ has a square root as its outermost operation. We are given $g(t) = \sqrt{t}$, which is a square root function. This means $g(t)$ is the outer function.

For $f(x) = g(h(x))$, we substitute $h(x)$ into $g(t)$.
Since $g(t) = \sqrt{t}$ and $f(x) = \sqrt{x^3 + 1}$, we can see that the expression inside the square root in $f(x)$ corresponds to $t$ in $g(t)$.
Therefore, the inner function $h(x)$ must be $x^3 + 1$.

The two functions are:

When composed, $g(h(x)) = g(x^3 + 1) = \sqrt{x^3 + 1}$, which is $f(x)$.

The two functions are $g(t) = \sqrt{t} \text{ and } h(x) = x^3 + 1$.