Determine whether the following functions are even or odd or neither even nor odd:
`x+x^(2)`

Explanation

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

### Step 1: Define the function
The given function is:
\[ f(x) = x + x^2 \]

### Step 2: Find \( f(-x) \)
To check if the function is even or odd, we need to calculate \( f(-x) \):
\[ f(-x) = -x + (-x)^2 \]
Calculating \( (-x)^2 \):
\[ (-x)^2 = x^2 \]
Thus,
\[ f(-x) = -x + x^2 \]

### Step 3: Check if the function is even
A function is even if:
\[ f(x) = f(-x) \]
We compare \( f(x) \) and \( f(-x) \):
\[ f(x) = x + x^2 \]
\[ f(-x) = -x + x^2 \]
Since \( f(x) \neq f(-x) \) (because \( x + x^2 \neq -x + x^2 \)), the function is not even.

### Step 4: Check if the function is odd
A function is odd if:
\[ f(-x) = -f(x) \]
Calculating \( -f(x) \):
\[ -f(x) = -(x + x^2) = -x - x^2 \]
Now we compare \( f(-x) \) and \( -f(x) \):
\[ f(-x) = -x + x^2 \]
\[ -f(x) = -x - x^2 \]
Since \( f(-x) \neq -f(x) \) (because \( -x + x^2 \neq -x - x^2 \)), the function is not odd.

### Conclusion
Since \( f(x) \) is neither even nor odd, we conclude:
\[ \text{The function } f(x) = x + x^2 \text{ is neither even nor odd.} \]

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