Given $f(x)=x^{3}+1$ and $g(x)=x-1$, simplify the function operation $f(x)-g(x)$

Step 1 ：Given the functions $f(x)=x^{3}+1$ and $g(x)=x-1$

Step 2 ：We need to subtract the function $g(x)$ from the function $f(x)$

Step 3 ：This means we need to subtract each term of $g(x)$ from the corresponding term of $f(x)$

Step 4 ：Subtracting $g(x)$ from $f(x)$ gives us $x^{3} - x + 2$

Step 5 ：So, the simplified form of the function operation $f(x)-g(x)$ is \(\boxed{x^{3} - x + 2}\)