Question

7
If : R-{0} Ris defined by f(x) = x + 1/x then
Prove that [f(x)]^2 = f(x^2)+ f(1)
​

Answer 1

Solution :

• Given : f(x) = x + 1/x
• To prove : [f(x)]² = f(x²) + f(1)

Proof :

We have ,

f(x) = x + 1/x

Now ,

=> LHS = [f(x)]²

=> LHS = (x + 1/x)²

=> LHS = x² + (1/x)² + 2•x•(1/x)

=> LHS = x² + 1/x² + 2

Also ,

=> RHS = f(x²) + f(1)

=> RHS = (x² + 1/x²) + (1 + 1/1)

=> RHS = x² + 1/x² + 1 + 1

=> RHS = x² + 1/x² + 2

Clearly ,

LHS = RHS

Hence proved .