We are given the functions $f(x) = 3x - 1$ and $g(x) = x^2 - 1$. We are tasked with finding $(f \circ g)(0)$, which is the composition of $f$ and $g$ at $x = 0$.

Step-by-step solution:

1. Composition of functions:
   $(f \circ g)(x) = f(g(x))$, meaning we first evaluate $g(x)$ and then substitute the result into $f(x)$.

2. Find $g(0)$:
   Since $g(x) = x^2 - 1$, substituting $0$ into $g(x)$: $g(0) = 0^2 - 1 = -1$

3. Evaluate $f(g(0))$:
   Now, we substitute $g(0) = -1$ into $f(x)$. Since $f(x) = 3x - 1$, we calculate: $f(-1) = 3(-1) - 1 = -3 - 1 = -4$

Thus, $(f \circ g)(0) = -4$.

Final Answer:

$(f \circ g)(0) = -4$

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:

1. What is $(g \circ f)(0)$ for the same functions $f(x) = 3x - 1$ and $g(x) = x^2 - 1$?
2. How would you find $(f \circ g)(1)$?
3. Can you compute $(f \circ f)(0)$?
4. What is the domain of $f \circ g$?
5. How does the composition of functions differ from addition of functions?

Tip: When working with compositions, always evaluate the inner function first!