Question

If \( f : \mathbb{R} \setminus \{0\} \to \mathbb{R} \) is defined by \( f(x) = x + \frac{1}{x} \), then the value of \( \left(f(x)\right)^2 \) is:

• A. \( f(x) + f(0) \)
• B. \( f(x^2) + f(2) \)
• C. \( f(x^3) + f(0) \)
• D. \( f(x^2) + f(1) \)

Hint:

When squaring a function like \( f(x) = x + \frac{1}{x} \), look for patterns that can be rewritten in terms of other function values to simplify the expression.

Answer 1

The Correct Option is D

Step 1: Use the definition of the function.
Given \( f(x) = x + \frac{1}{x} \), we square it: \[ f(x)^2 = \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} \]
Step 2: Rewrite the RHS using values of \( f(x^2) \) and \( f(1) \). \[ f(x^2) = x^2 + \frac{1}{x^2}, \quad f(1) = 1 + \frac{1}{1} = 2 \] \[ \Rightarrow f(x)^2 = f(x^2) + f(1) \]