Determine whether function, `f(x)=(-1)^([x])` is even, odd or neither of two (where `[*]` denotes the greatest integer function).

Answer

To determine whether the function \( f(x) = (-1)^{[x]} \) (where \([x]\) denotes the greatest integer function) is even, odd, or neither, we will follow these steps:

### Step 1: Understand the Definitions
A function \( f(x) \) is:
- **Even** if \( f(x) = f(-x) \) for all \( x \).
- **Odd** if \( f(x) = -f(-x) \) for all \( x \).
- **Neither** if it does not satisfy either condition.

### Step 2: Evaluate \( f(x) \) and \( f(-x) \)
We need to evaluate \( f(x) \) and \( f(-x) \) for various values of \( x \).

1. **For \( x = 0 \)**:
- \( f(0) = (-1)^{[0]} = (-1)^0 = 1 \)
- \( f(-0) = f(0) = 1 \)

2. **For \( x = 0.5 \)**:
- \( f(0.5) = (-1)^{[0.5]} = (-1)^0 = 1 \)
- \( f(-0.5) = (-1)^{[-0.5]} = (-1)^{-1} = -1 \)

3. **For \( x = 1 \)**:
- \( f(1) = (-1)^{[1]} = (-1)^1 = -1 \)
- \( f(-1) = (-1)^{[-1]} = (-1)^{-1} = -1 \)

4. **For \( x = 1.5 \)**:
- \( f(1.5) = (-1)^{[1.5]} = (-1)^1 = -1 \)
- \( f(-1.5) = (-1)^{[-1.5]} = (-1)^{-2} = 1 \)

5. **For \( x = 2 \)**:
- \( f(2) = (-1)^{[2]} = (-1)^2 = 1 \)
- \( f(-2) = (-1)^{[-2]} = (-1)^{-2} = 1 \)

### Step 3: Compare \( f(x) \) and \( f(-x) \)
Now we compare the values we calculated:

- For \( x = 0 \): \( f(0) = f(-0) \) (even)
- For \( x = 0.5 \): \( f(0.5) \neq f(-0.5) \) and \( f(0.5) \neq -f(-0.5) \) (neither)
- For \( x = 1 \): \( f(1) = f(-1) \) (even)
- For \( x = 1.5 \): \( f(1.5) \neq f(-1.5) \) and \( f(1.5) \neq -f(-1.5) \) (neither)
- For \( x = 2 \): \( f(2) = f(-2) \) (even)

### Step 4: Conclusion
Since the function does not consistently satisfy the conditions for being even or odd across all values of \( x \), we conclude that the function \( f(x) = (-1)^{[x]} \) is **neither even nor odd**.