Question

if f(x)= x-1/x prove that [f(x)]^3=f(x)^3 -3f(1/x)

Answer 1

First of all the question should be,
Prove that
f(x³)=[f(x)]³-3f(1/x)

So,
First considering Right hand side,

=[f(x)]³-3f(1/x)
=(x- 1/x )³ -3 (1/x - x)
=(x³ - (1/x)³)  - 3(x × (1/x) (x- 1/x)) -3 (1/x - x)
=(x³ - (1/x)³)  -3 (x- 1/x)   +3 ( x - 1/x)
=x³ - (1/x)³

Now, considering Left hand side.
f(x³)=x³ - (1/x)³

So,
We conclude that LHS= RHS
Hence proved

Answer 2

In the attachments I have answered this problem.

I have applied the following algebraic identity to get the required result.

(a-b)^3 = a^3 - b^3 - 3ab(a-b)

The above identity is modified and suitable simplifications are made to get the final answer.

I hope this answer helps you