Let `f(x+(1)/(x))=x^(2)+(1)/(x^(2)),(xne0)`, then `f(x)` is equal to :

Options

A. `x^(2)-1`
B. `x^(2)-2`
C. `x^(2)`
D. none of these

Answer

To solve the problem, we start with the given equation:

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

### Step 1: Rewrite \( x^2 + \frac{1}{x^2} \)
We can use the identity for \( (a + b)^2 \):

\[ a^2 + b^2 = (a + b)^2 - 2ab \]

Let \( a = x \) and \( b = \frac{1}{x} \). Therefore:

\[ x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2 \]

### Step 2: Substitute into the function
Now we can substitute this back into our original equation:

\[ f\left(x + \frac{1}{x}\right) = \left(x + \frac{1}{x}\right)^2 - 2 \]

### Step 3: Define a new variable
Let:

\[ y = x + \frac{1}{x} \]

Then we can rewrite the function as:

\[ f(y) = y^2 - 2 \]

### Step 4: Substitute back to find \( f(x) \)
Since \( y = x + \frac{1}{x} \), we want to express \( f(x) \) in terms of \( x \). We already have:

\[ f\left(x + \frac{1}{x}\right) = \left(x + \frac{1}{x}\right)^2 - 2 \]

Thus, we can express \( f(x) \) as:

\[ f(x) = x^2 - 2 \]

### Conclusion
Therefore, the function \( f(x) \) is:

\[ f(x) = x^2 - 2 \]

### Final Answer
The correct option is \( \text{B: } x^2 - 2 \).

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