Question

If the function f :R–{1, –1}→A defined by f (x) = x²/(1 - x²), is surjective, then A is equal to:
(A) R – [–1,0) (B) R–(–1,0)
(C) R–{–1} (D) [0, [infinity]]

Answer 1

A is equal to R – [–1 , 0 )

Given function f :R–{1, –1}→A defined by f ( x ) = x²/(1 - x²)

• Let f ( x ) = y = x²/(1 - x²)

Now solving for x²,

• x² = y ( 1 - x² )

• x² = y - yx²

• x² + yx² = y

• x² = y / ( 1 + y )

we know,

• x²  ≥ 0
• y / ( 1 + y ) ≥ 0
• y ∈ ( - ∞ , - 1 ) ∪ [ 0 , ∞ )

We know that,

• For surjective function, Range  = codomain,

• A = ( - ∞ , - 1 ) ∪ [ 0 , ∞ )
• A = R - [ - 1 , 0 )