If `f(x + 1/x) = x^2 + 1/(x^2)` , then f(t) equals to :

Options

A. `t^2 - 1/(t^2)`
B. `t^2 + 1/(t^2)`
C. `t^2 + 2`
D. `(t - 1)^2`

Answer

To solve the problem, we need to find the function \( f(t) \) given that \( f(x + \frac{1}{x}) = x^2 + \frac{1}{x^2} \).

### Step-by-Step Solution:

1. **Substitution**: We start by letting \( t = x + \frac{1}{x} \). This means we want to express \( f(t) \) in terms of \( t \).

**Hint**: Identify the relationship between \( t \) and \( x \) to simplify the expression.

2. **Square Both Sides**: Next, we square both sides of the equation \( t = x + \frac{1}{x} \):
\[ t^2 = \left(x + \frac{1}{x}\right)^2 \]
Expanding the right-hand side, we get:
\[ t^2 = x^2 + 2 + \frac{1}{x^2} \]
Rearranging gives:
\[ t^2 = x^2 + \frac{1}{x^2} + 2 \]

**Hint**: Remember that squaring a binomial involves adding the square of each term and twice the product of the terms.

3. **Isolate \( x^2 + \frac{1}{x^2} \)**: From the equation \( t^2 = x^2 + \frac{1}{x^2} + 2 \), we can isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = t^2 - 2 \]

**Hint**: Rearranging equations can help isolate the terms you need.

4. **Substitute Back into the Function**: Now that we have \( x^2 + \frac{1}{x^2} \) in terms of \( t \), we can substitute this back into the original function:
\[ f(t) = x^2 + \frac{1}{x^2} = t^2 - 2 \]

**Hint**: Make sure to replace the variable in the function with the new variable you defined.

5. **Final Answer**: Therefore, the function \( f(t) \) is:
\[ f(t) = t^2 - 2 \]

### Summary:
The final result is:
\[ f(t) = t^2 - 2 \]