`f:Nto[-sqrt2,sqrt2]:f(x)=sinx+cosx`, then f(x) is

Options

A. one-one.
B. onto
C. many-one
D. into

Answer

To solve the problem, we need to analyze the function \( f(x) = \sin x + \cos x \) with the given domain \( \mathbb{N} \) (natural numbers) and co-domain \( [-\sqrt{2}, \sqrt{2}] \).

### Step 1: Rewrite the Function
We can rewrite the function \( f(x) \) using the sine addition formula.

\[ f(x) = \sin x + \cos x = \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x \right) \]

Using the sine addition formula \( \sin(a + b) = \sin a \cos b + \cos a \sin b \), we can express this as:

\[ f(x) = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right) \]

### Step 2: Determine the Range of \( f(x) \)
The sine function oscillates between -1 and 1. Therefore, the maximum and minimum values of \( f(x) \) can be calculated as follows:

- Maximum value:
\[ \sqrt{2} \cdot 1 = \sqrt{2} \]
- Minimum value:
\[ \sqrt{2} \cdot (-1) = -\sqrt{2} \]

Thus, the range of \( f(x) \) is \( [-\sqrt{2}, \sqrt{2}] \).

### Step 3: Analyze the Domain
The domain of \( f(x) \) is restricted to natural numbers \( \mathbb{N} \). Since \( x \) can only take values like 1, 2, 3, etc., we need to evaluate \( f(x) \) at these points.

### Step 4: Check Values of \( f(x) \) at Natural Numbers
Let's calculate \( f(x) \) for a few natural numbers:

- \( f(1) = \sin(1) + \cos(1) \)
- \( f(2) = \sin(2) + \cos(2) \)
- \( f(3) = \sin(3) + \cos(3) \)

These values will lie within the range \( [-\sqrt{2}, \sqrt{2}] \).

### Step 5: Determine if the Function is One-One
To check if \( f(x) \) is one-one, we need to see if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \).

Assume \( f(x_1) = f(x_2) \):
\[ \sqrt{2} \sin \left( x_1 + \frac{\pi}{4} \right) = \sqrt{2} \sin \left( x_2 + \frac{\pi}{4} \right) \]

This implies:
\[ \sin \left( x_1 + \frac{\pi}{4} \right) = \sin \left( x_2 + \frac{\pi}{4} \right) \]

The sine function is periodic, but since \( x_1 \) and \( x_2 \) are restricted to natural numbers, the only solution is \( x_1 = x_2 \). Therefore, \( f(x) \) is one-one.

### Conclusion
The function \( f(x) = \sin x + \cos x \) has a range of \( [-\sqrt{2}, \sqrt{2}] \) and is one-one when defined over the natural numbers.