Examine whether the following function are even or odd or neither even nor odd, where [ ] denotes greatest integer function.
`f(x)=sqrt(1+x+x^(2))-sqrt(1-x+x^(2))`

Explanation

To determine whether the function \( f(x) = \sqrt{1 + x + x^2} - \sqrt{1 - x + x^2} \) is even, odd, or neither, we will follow these steps:

### Step 1: Define the function
We have:
\[ f(x) = \sqrt{1 + x + x^2} - \sqrt{1 - x + x^2} \]

### Step 2: Find \( f(-x) \)
To check if the function is even or odd, we need to compute \( f(-x) \):
\[ f(-x) = \sqrt{1 + (-x) + (-x)^2} - \sqrt{1 - (-x) + (-x)^2} \]
This simplifies to:
\[ f(-x) = \sqrt{1 - x + x^2} - \sqrt{1 + x + x^2} \]

### Step 3: Compare \( f(-x) \) with \( f(x) \)
Now we will compare \( f(-x) \) with \( f(x) \):
\[ f(-x) = -\left(\sqrt{1 + x + x^2} - \sqrt{1 - x + x^2}\right) = -f(x) \]
This shows that:
\[ f(-x) = -f(x) \]

### Step 4: Conclusion
Since \( f(-x) = -f(x) \), we conclude that the function \( f(x) \) is an **odd function**.

### Summary
- \( f(x) \) is odd because \( f(-x) = -f(x) \).

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