Question

If \( f(x) = x + \frac{1}{x} \), prove that \( \left[ f(x) \right]^3 = f(x^3) + 3f\left( \frac{1}{x} \right) \).

Hint:

For proving identities involving functions, expand powers and substitute the function's value at different points.

Answer 1

Given that \( f(x) = x + \frac{1}{x} \), we want to prove the following: \[ \left( f(x) \right)^3 = f(x^3) + 3f\left( \frac{1}{x} \right) \] First, calculate \( f(x)^3 \): \[ f(x) = x + \frac{1}{x} \] \[ f(x)^3 = \left( x + \frac{1}{x} \right)^3 = x^3 + 3x + 3\frac{1}{x} + \frac{1}{x^3} \] Now, calculate \( f(x^3) \) and \( f\left( \frac{1}{x} \right) \): \[ f(x^3) = x^3 + \frac{1}{x^3} \] \[ f\left( \frac{1}{x} \right) = \frac{1}{x} + x \] Now, substitute these into the right-hand side of the equation: \[ f(x^3) + 3f\left( \frac{1}{x} \right) = \left( x^3 + \frac{1}{x^3} \right) + 3 \left( \frac{1}{x} + x \right) \] Simplifying the right-hand side: \[ = x^3 + \frac{1}{x^3} + 3x + \frac{3}{x} \] This is exactly the same as \( f(x)^3 \).
Final Answer:
Thus, we have proven that: \[ f(x)^3 = f(x^3) + 3f\left( \frac{1}{x} \right) \]