Determine whether the following functions are even or odd or neither even nor odd:
`sin(x^(2)+1)`

Explanation

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

### Step 1: Define the function
Let \( f(x) = \sin(x^2 + 1) \).

### Step 2: Find \( f(-x) \)
To check if the function is even or odd, we need to compute \( f(-x) \):
\[ f(-x) = \sin((-x)^2 + 1) \]
Since \((-x)^2 = x^2\), we have:
\[ f(-x) = \sin(x^2 + 1) \]

### Step 3: Compare \( f(-x) \) with \( f(x) \)
Now we compare \( f(-x) \) with \( f(x) \):
\[ f(x) = \sin(x^2 + 1) \]
\[ f(-x) = \sin(x^2 + 1) \]
Since \( f(-x) = f(x) \), this means the function satisfies the condition for being an even function.

### Conclusion
Thus, the function \( f(x) = \sin(x^2 + 1) \) is an **even function**.

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