Which of the following functions is (are) even, odd or neither:
`f(x)=sqrt(1+x+x^(2))-sqrt(1-x+x^(2))`

Answer

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: Find \( f(-x) \)
We start by substituting \(-x\) into the function:

\[ f(-x) = \sqrt{1 + (-x) + (-x)^2} - \sqrt{1 - (-x) + (-x)^2} \]

This simplifies to:

\[ f(-x) = \sqrt{1 - x + x^2} - \sqrt{1 + x + x^2} \]

### Step 2: Rearranging \( f(-x) \)
Now, we can rearrange the expression for \( f(-x) \):

\[ f(-x) = -\left(\sqrt{1 + x + x^2} - \sqrt{1 - x + x^2}\right) \]

This can be rewritten as:

\[ f(-x) = -f(x) \]

### Step 3: Conclusion
Since we have found that \( f(-x) = -f(x) \), we can conclude that the function \( f(x) \) is an odd function.

### Final Answer
The function \( f(x) = \sqrt{1+x+x^2} - \sqrt{1-x+x^2} \) is an **odd function**.

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