The range of g(x) =[[x/2]] is:

To determine the range of $g(x) = \left\lfloor \frac{x}{2} \right\rfloor$, follow these steps:

1. Understand the function: The function $g(x) = \left\lfloor \frac{x}{2} \right\rfloor$ involves the floor function, which returns the greatest integer less than or equal to the input.

2. Analyze the input: For any real number $x$, $\frac{x}{2}$ can be any real number. The floor function will then map this real number to the nearest integer less than or equal to it.

3. Determine the output: Since $\frac{x}{2}$ can be any real number, the floor function will produce all possible integers. For example:

   • If $x = 0$, $g(x) = \left\lfloor 0 \right\rfloor = 0$.
   • If $x = 1$, $g(x) = \left\lfloor 0.5 \right\rfloor = 0$.
   • If $x = 2$, $g(x) = \left\lfloor 1 \right\rfloor = 1$.
   • If $x = -1$, $g(x) = \left\lfloor -0.5 \right\rfloor = -1$.
   • If $x = -2$, $g(x) = \left\lfloor -1 \right\rfloor = -1$.

4. Identify the range: Since $\frac{x}{2}$ can be any real number, and the floor function maps any real number to an integer, the range of $g(x)$ is all integers.

Therefore, the range of $g(x) = \left\lfloor \frac{x}{2} \right\rfloor$ is: $\text{Range} = \mathbb{Z}$ where $\mathbb{Z}$ denotes the set of all integers.