To find $g(1)$, we substitute $x = 1$ into the function $g(x) = 4|x - 3| + 2$.

Step-by-step solution:

1. Start with the function: $g(x) = 4|x - 3| + 2$
2. Substitute $x = 1$: $g(1) = 4|1 - 3| + 2$
3. Simplify inside the absolute value: $g(1) = 4| -2 | + 2$
4. The absolute value of $-2$ is $2$, so: $g(1) = 4(2) + 2$
5. Multiply: $g(1) = 8 + 2$
6. Add: $g(1) = 10$

Thus, $g(1) = 10$.

Would you like more details on this problem, or do you have any questions?

Here are 5 related questions:

1. What is $g(3)$ for the same function?
2. How would the graph of $g(x) = 4|x - 3| + 2$ look?
3. What happens to the function when $x > 3$ vs. $x < 3$?
4. How would you solve $g(x) = 14$ for $x$?
5. What is the vertex of the function $g(x)$ in terms of transformations?

Tip: The graph of an absolute value function $|x - a|$ forms a "V" shape with the vertex at $x = a$.