Question 1209611
Here's how to determine the domain of g:

1. **Understand the relationship between f and g:**

The function g(x) is defined in terms of f. The input to f is the expression x - 4x². For g(x) to be defined, this expression must fall within the domain of f, which is (-1, 3).

2. **Set up the inequality:**

We need to find the values of x for which:

-1 < x - 4x² < 3

3. **Solve the inequalities:**

We'll solve this as two separate inequalities:

* **-1 < x - 4x²**
   4x² - x - 1 < 0

   Find the roots of 4x² - x - 1 = 0 using the quadratic formula:

   x = (1 ± √(1 + 16))/8 = (1 ± √17)/8

   So, (1 - √17)/8 < x < (1 + √17)/8

* **x - 4x² < 3**
   4x² - x + 3 > 0

   Find the roots of 4x² - x + 3 = 0. The discriminant is 1 - 4*4*3 = -47, which is negative.  This means the quadratic is always positive, so this inequality is true for all x.

4. **Combine the solutions:**

Since the second inequality is true for all x, we only need to consider the first inequality.

5. **State the domain of g:**

The domain of g is:

((1 - √17)/8, (1 + √17)/8)