The problem asks us to determine whether the function $f(x)$ defined as $f(x) = \begin{cases} x+5, & -2 < x < 0 \\ -x+5, & 0 \le x < 2 \end{cases}$ is an odd function, an even function, neither odd nor even, or none of the above.

1. Problem Description

The problem asks us to determine whether the function $f(x)$ defined as

$f(x) = \begin{cases} x+5, & -2 < x < 0 \\ -x+5, & 0 \le x < 2 \end{cases}$

is an odd function, an even function, neither odd nor even, or none of the above.

2. Solution Steps

To determine whether a function is even, we check if $f(-x) = f(x)$ for all $x$ in the domain. To determine whether a function is odd, we check if $f(-x) = -f(x)$ for all $x$ in the domain.

First, let's find $f(-x)$ for $-2 < x < 0$. In this case, $0 < -x < 2$, so we use the second case in the definition of $f(x)$:

$f(-x) = -(-x) + 5 = x+5$.

Now, let's find $f(-x)$ for $0 \le x < 2$. In this case, $-2 < -x \le 0$. If $-x=0$, then $x=0$. If $-2 < -x < 0$, we use the first case in the definition of $f(x)$:

$f(-x) = (-x)+5 = -x+5$.

Thus,

$f(-x) = \begin{cases} -x+5, & -2 < -x < 0 \\ -(-x)+5, & 0 \le -x < 2 \end{cases} = \begin{cases} -x+5, & 0 < x < 2 \\ x+5, & -2 < x \le 0 \end{cases}$

Comparing $f(-x)$ with $f(x)$, we see that

$f(x) = \begin{cases} x+5, & -2 < x < 0 \\ -x+5, & 0 \le x < 2 \end{cases}$

$f(-x) = \begin{cases} -x+5, & 0 < x < 2 \\ x+5, & -2 < x \le 0 \end{cases}$

If $f(x)$ is even, then $f(x) = f(-x)$. This means

$x+5 = -x+5$ for $0 < x < 2$, and

$-x+5 = x+5$ for $-2 < x < 0$.

These conditions simplify to $x=0$ in both cases, which is not true for all $x$ in the intervals. Therefore, $f(x)$ is not an even function.

If $f(x)$ is odd, then $f(-x) = -f(x)$. This means

$-f(x) = \begin{cases} -(x+5), & -2 < x < 0 \\ -(-x+5), & 0 \le x < 2 \end{cases} = \begin{cases} -x-5, & -2 < x < 0 \\ x-5, & 0 \le x < 2 \end{cases}$

Since $f(-x) = \begin{cases} -x+5, & 0 < x < 2 \\ x+5, & -2 < x \le 0 \end{cases}$, for $f(x)$ to be odd,

we require $x+5 = -x-5$ for $-2 < x < 0$, which means $2x = -10$, so $x=-5$. This is outside the given interval, so the function is not odd in this interval.

Also, we require $-x+5 = x-5$ for $0 \le x < 2$, which means $2x=10$, so $x=5$. This is outside the given interval, so the function is not odd in this interval.

Therefore, $f(x)$ is not an odd function.

Since $f(x)$ is neither even nor odd, the answer is (c).

3. Final Answer

(c) Neither odd nor even