Find the range of the following functions: (where {.} and [.] represent fractional part and greatest integer part functions respectively)
`f(x)=x^(3)-12x` where `xepsilon[-3,1]`

Explanation

To find the range of the function \( f(x) = x^3 - 12x \) for \( x \in [-3, 1] \), we will follow these steps:

### Step 1: Identify the function and its domain
The function given is \( f(x) = x^3 - 12x \) and the domain is \( x \in [-3, 1] \).

### Step 2: Find the critical points
To find the critical points, we need to differentiate the function and set the derivative equal to zero.

\[ f'(x) = 3x^2 - 12 \]
Setting the derivative to zero:
\[ 3x^2 - 12 = 0 \]
\[ 3x^2 = 12 \]
\[ x^2 = 4 \]
\[ x = \pm 2 \]

### Step 3: Determine which critical points are within the domain
The critical points we found are \( x = 2 \) and \( x = -2 \). Since our domain is \( [-3, 1] \), only \( x = -2 \) is relevant.

### Step 4: Evaluate the function at the endpoints and the critical point
Next, we will evaluate \( f(x) \) at the endpoints of the interval and at the critical point \( x = -2 \).

1. **At \( x = -3 \)**:
\[ f(-3) = (-3)^3 - 12(-3) = -27 + 36 = 9 \]

2. **At \( x = -2 \)**:
\[ f(-2) = (-2)^3 - 12(-2) = -8 + 24 = 16 \]

3. **At \( x = 1 \)**:
\[ f(1) = (1)^3 - 12(1) = 1 - 12 = -11 \]

### Step 5: Determine the range
Now we compare the values obtained:
- \( f(-3) = 9 \)
- \( f(-2) = 16 \)
- \( f(1) = -11 \)

The minimum value is \( -11 \) and the maximum value is \( 16 \).

### Conclusion
Thus, the range of the function \( f(x) = x^3 - 12x \) for \( x \in [-3, 1] \) is:

\[ \text{Range} = [-11, 16] \]