Let `f(x+(1)/(x))=x^(2)+(1)/(x^(2)),(x ne 0)` then f(x) equals

Options

A. `x^(2)-` for all x
B. `x^(2)-2"for all" |x| gt 2`
C. `x^(2)-2"for all "|x|lt2`
D. none of these

Answer

To solve the problem, we need to find the function \( f(x) \) given that:

\[ f\left(x + \frac{1}{x}\right) = x^2 + \frac{1}{x^2}, \quad (x \neq 0) \]

### Step 1: Rewrite the Right Side
We know the identity for \( a^2 + b^2 \):
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
In our case, let \( a = x \) and \( b = \frac{1}{x} \). Then:
\[ x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2 \cdot x \cdot \frac{1}{x} \]
This simplifies to:
\[ x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2 \]

### Step 2: Substitute Back into the Function
Now, substituting this back into the function:
\[ f\left(x + \frac{1}{x}\right) = \left(x + \frac{1}{x}\right)^2 - 2 \]

### Step 3: Define \( y = x + \frac{1}{x} \)
Let \( y = x + \frac{1}{x} \). Then we can express \( f(y) \):
\[ f(y) = y^2 - 2 \]

### Step 4: Conclusion
Thus, the function \( f(x) \) can be expressed as:
\[ f(x) = x^2 - 2 \]

### Final Answer
The function \( f(x) \) is:
\[ \boxed{x^2 - 2} \]