To find the value of $g(f(0))$, we first need to evaluate $f(0)$ and then substitute that result into $g(x)$. The steps are as follows:

Step 1: Find $f(0)$

The function $f(x)$ is given by: $f(x) = 3x + 1$ Substitute $x = 0$ into $f(x)$: $f(0) = 3(0) + 1 = 1$

Step 2: Find $g(f(0)) = g(1)$

The function $g(x)$ is given by: $g(x) = x^2 - 2x - 6$ Now substitute $x = 1$ into $g(x)$: $g(1) = (1)^2 - 2(1) - 6 = 1 - 2 - 6 = -7$

Final Answer:

$g(f(0)) = -7$

Would you like more details on any step? Here are five related questions for practice:

1. What is $f(2)$ using the same function $f(x) = 3x + 1$?
2. What is $g(2)$ for the given function $g(x) = x^2 - 2x - 6$?
3. What is $g(f(2))$?
4. If $h(x) = 2x + 3$, what is $h(f(1))$?
5. Can you find the value of $f(g(1))$?

Tip: When dealing with composite functions like $g(f(x))$, always evaluate the inner function first before substituting its result into the outer function.