To solve this problem, we need to substitute $x - 3$ into the given function $g(x) = 3x + 1$.

Here's the step-by-step process:

1. The function $g(x)$ is defined as: $g(x) = 3x + 1$

2. We are asked to find $g(x - 3)$, so we substitute $x - 3$ in place of $x$ in the function: $g(x - 3) = 3(x - 3) + 1$

3. Simplify the expression: $g(x - 3) = 3x - 9 + 1 = 3x - 8$

Thus, the solution is: $g(x - 3) = 3x - 8$

Would you like any further explanation on this?

Here are some related questions that might help expand your understanding:

1. What would $g(x+2)$ be for the same function $g(x) = 3x + 1$?
2. How would you find $g(2x)$ using the given function definition?
3. What is the inverse of the function $g(x) = 3x + 1$?
4. How do you determine the domain and range of the function $g(x) = 3x + 1$?
5. What would be the result of solving $g(x - a)$ in terms of $a$?

Tip: When substituting into functions, carefully replace every occurrence of the variable with the new expression to avoid errors.