Determine whether the following functions are even or odd or neither even nor odd:
`f(x)=sinx+cosx`

Explanation

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

### Step 1: Recall the definitions of even and odd functions
- A function \( f(x) \) is **even** if \( f(-x) = f(x) \) for all \( x \).
- A function \( f(x) \) is **odd** if \( f(-x) = -f(x) \) for all \( x \).

### Step 2: Find \( f(-x) \)
To find \( f(-x) \), we substitute \(-x\) into the function:

\[ f(-x) = \sin(-x) + \cos(-x) \]

### Step 3: Use the properties of sine and cosine
Using the properties of sine and cosine:
- \( \sin(-x) = -\sin(x) \)
- \( \cos(-x) = \cos(x) \)

Substituting these into our expression for \( f(-x) \):

\[ f(-x) = -\sin(x) + \cos(x) \]

### Step 4: Compare \( f(-x) \) with \( f(x) \) and \(-f(x)\)
Now we will compare \( f(-x) \) with \( f(x) \) and \(-f(x)\):

1. **Check if \( f(-x) = f(x) \)**:
\[ f(x) = \sin(x) + \cos(x) \]
Since \( f(-x) = -\sin(x) + \cos(x) \), we see that \( f(-x) \neq f(x) \).

2. **Check if \( f(-x) = -f(x) \)**:
\[ -f(x) = -(\sin(x) + \cos(x)) = -\sin(x) - \cos(x) \]
Since \( f(-x) = -\sin(x) + \cos(x) \), we see that \( f(-x) \neq -f(x) \).

### Step 5: Conclusion
Since \( f(-x) \) is neither equal to \( f(x) \) nor equal to \(-f(x)\), we conclude that the function \( f(x) = \sin x + \cos x \) is neither even nor odd.

### Summary of Steps
1. Recall definitions of even and odd functions.
2. Find \( f(-x) \).
3. Use properties of sine and cosine to simplify \( f(-x) \).
4. Compare \( f(-x) \) with \( f(x) \) and \(-f(x)\).
5. Conclude that the function is neither even nor odd.