Question 1209616
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 + 1)(x - 2).  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 + 1)(x - 2) < 3

**3. Solve the inequalities:**

We'll solve this as two separate inequalities:

* **-1 < (x + 1)(x - 2)**
   -1 < x² - x - 2
   0 < x² - x - 1

   To solve this, we find the roots of the quadratic x² - x - 1 = 0 using the quadratic formula:

   x = (1 ± √(1 + 4))/2 = (1 ± √5)/2

   So, x < (1 - √5)/2 or x > (1 + √5)/2

* **(x + 1)(x - 2) < 3**
   x² - x - 2 < 3
   x² - x - 5 < 0

   Again, find the roots of x² - x - 5 = 0:

   x = (1 ± √(1 + 20))/2 = (1 ± √21)/2

   So, (1 - √21)/2 < x < (1 + √21)/2

**4. Combine the solutions:**

We need to find where *both* inequalities are true.  Let's approximate the roots:

* (1 - √5)/2 ≈ -0.618
* (1 + √5)/2 ≈ 1.618
* (1 - √21)/2 ≈ -1.791
* (1 + √21)/2 ≈ 2.791

Combining the inequalities, we get:

(1 - √21)/2 < x < (1 - √5)/2  *or*  (1 + √5)/2 < x < (1 + √21)/2

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

The domain of g is the union of the two intervals:

((1 - √21)/2, (1 - √5)/2) ∪ ((1 + √5)/2, (1 + √21)/2)