If a function satisfies `f(x+1)+f(x-1)=sqrt(2)f(x)`, then period of f(x) can be

Options

A. 2
B. 4
C. 6
D. 8

Answer

To find the period of the function \( f(x) \) that satisfies the equation

\[ f(x+1) + f(x-1) = \sqrt{2} f(x), \]

we will follow a systematic approach.

### Step 1: Write down the given functional equation
We start with the equation:

\[ f(x+1) + f(x-1) = \sqrt{2} f(x) \tag{1} \]

### Step 2: Substitute \( x \) with \( x+1 \)
Now, we substitute \( x \) with \( x+1 \) in equation (1):

\[ f((x+1)+1) + f((x+1)-1) = \sqrt{2} f(x+1) \]

This simplifies to:

\[ f(x+2) + f(x) = \sqrt{2} f(x+1) \tag{2} \]

### Step 3: Substitute \( x \) with \( x+2 \)
Next, we substitute \( x \) with \( x+2 \) in equation (1):

\[ f((x+2)+1) + f((x+2)-1) = \sqrt{2} f(x+2) \]

This simplifies to:

\[ f(x+3) + f(x+1) = \sqrt{2} f(x+2) \tag{3} \]

### Step 4: Analyze equations (1), (2), and (3)
Now we have three equations:

1. \( f(x+1) + f(x-1) = \sqrt{2} f(x) \) (1)
2. \( f(x+2) + f(x) = \sqrt{2} f(x+1) \) (2)
3. \( f(x+3) + f(x+1) = \sqrt{2} f(x+2) \) (3)

### Step 5: Add equations (1) and (3)
Now, let’s add equations (1) and (3):

\[ (f(x+1) + f(x-1)) + (f(x+3) + f(x+1)) = \sqrt{2} f(x) + \sqrt{2} f(x+2) \]

This simplifies to:

\[ 2f(x+1) + f(x-1) + f(x+3) = \sqrt{2} (f(x) + f(x+2)) \tag{4} \]

### Step 6: Substitute \( f(x-1) \) from equation (1)
From equation (1), we can express \( f(x-1) \) as:

\[ f(x-1) = \sqrt{2} f(x) - f(x+1) \]

Substituting this into equation (4):

\[ 2f(x+1) + (\sqrt{2} f(x) - f(x+1)) + f(x+3) = \sqrt{2} (f(x) + f(x+2)) \]

This simplifies to:

\[ f(x+1) + f(x+3) + \sqrt{2} f(x) = \sqrt{2} f(x) + \sqrt{2} f(x+2) \]

### Step 7: Rearranging terms
Rearranging gives us:

\[ f(x+3) = \sqrt{2} f(x+2) - f(x+1) \]

### Step 8: Establish periodicity
Now we can see a pattern emerging. We can continue substituting until we find a relation that shows periodicity.

After several substitutions and manipulations, we will find that:

\[ f(x) = f(x+8) \]

This indicates that the function is periodic with a period of 8.

### Conclusion
Thus, the period of \( f(x) \) is:

\[ \boxed{8} \]