Find $(f \circ g)(0)$ and $(g \circ f)(0)$.
\[
f(x)=2 x-1 ; g(x)=x^{2}-1
\]
\[
(f \circ g)(0)=
\]

Step 1 ：The question is asking for the composition of two functions, $f(x)$ and $g(x)$, evaluated at $x=0$. The composition of functions is a concept in mathematics where you apply one function to the result of another function. In this case, we are asked to find $(f \circ g)(0)$ and $(g \circ f)(0)$.

Step 2 ：For $(f \circ g)(0)$, this means we apply the function $g(x)$ first and then apply the function $f(x)$ to the result. So, we substitute $g(0)$ into $f(x)$.

Step 3 ：For $(g \circ f)(0)$, this means we apply the function $f(x)$ first and then apply the function $g(x)$ to the result. So, we substitute $f(0)$ into $g(x)$.

Step 4 ：Given that $f(x) = 2x - 1$ and $g(x) = x^{2} - 1$, we can substitute $g(0)$ into $f(x)$ to get $(f \circ g)(0) = 2(g(0)) - 1$.

Step 5 ：We know that $g(0) = 0^{2} - 1 = -1$, so we substitute $-1$ into the equation to get $(f \circ g)(0) = 2(-1) - 1 = -3$.

Step 6 ：So, the final answer for $(f \circ g)(0)$ is $\boxed{-3}$.

Step 7 ：For $(g \circ f)(0)$, we substitute $f(0)$ into $g(x)$ to get $(g \circ f)(0) = (f(0))^{2} - 1$.

Step 8 ：We know that $f(0) = 2(0) - 1 = -1$, so we substitute $-1$ into the equation to get $(g \circ f)(0) = (-1)^{2} - 1 = 0$.

Step 9 ：So, the final answer for $(g \circ f)(0)$ is $\boxed{0}$.