The domain of the function is the real numbers. However, there appears to be an interval where the function displays unexpected behavior. Around the interval between -2 and 2, the function changes its behavior. Before this interval, the output values are negative and gradually approach 0. After this interval, the output values become positive and continue to increase at a faster rate as the input value increases. The bounds of this interval are -2 and 2.

[CODE]:
```python
def f(x):
    if abs(x) < 2:
        return -0.5 * x ** 2 + 5.3
    else:
        return x ** 2 + 3.2 * x + 0.5
```

This code approximates the behavior of the function. It uses a piecewise function to capture the different behaviors before and after the interval. For input values within the interval, it uses a quadratic function with a negative coefficient, and for input values outside the interval, it uses a quadratic function with positive coefficients.