JEE Main2017MathematicsFunctionsActual
The function f : R → - 1 2 , 1 2 defined as f x = x 1 + x 2 , is:
Options
- AInvertible
- BInjective but not surjective
- CSurjective but not injective
- DNeither injective nor surjective
Correct answer
C. Surjective but not injective
Step-by-step solution
Let, y = x 1 + x 2 ⇒ y + y x 2 - x = 0 ⇒ x 2 y - x + y = 0 ⇒ x = + 1 ±     1 - 4 y 2 2 y ⇒ 1 - 4 y 2   ≥ 0 ⇒ 4 y 2   ≤ 1 ⇒ So, y ∈ - 1 2 , 1 2 Here, Co-Domain = Range Hence, the function is surjective. Now, for injective function ⇒ x 1 1 + x 1 2 = x 2 1 + x 2 2 ⇒ x 1 + x 2 x 2 2 - x 2 - x 2 x 1 2 = 0 ⇒ x 1 - x 2 + x 1 x 2 x 2 - x 1 = 0 ⇒ x 2 - x 1   x 1 x 2 - 1 = 0 ⇒ x 2 = x 1 or x 1 = 1 x 2 Hence, the