Question

The function f:R → R defined by f(x) = \(\rm \frac{x}{x^2+1}\), x ∈ R, is:
1. One to One but NOT Onto.
2. One to One and Onto.
3. NEITHER One to One NOR Onto.
4. None of these.

Answer 1

Correct Answer - Option 3 : NEITHER One to One NOR Onto.

Concept:

• One-One/Injective function: is a function that maps distinct elements of its domain to distinct elements of its range.
• Onto/Surjective function: is function in which every element of the domain has at least one image in the domain.
• One-One Onto/Bijective function: is a function which is both one-one and onto. i.e. each element of domain is paired with exactly one element of the range, and each element of the range is paired with exactly one element of domain.

 

Calculation:

For a and b ∈ R, consider f(a) = f(b).

⇒ \(\rm \frac{a}{a^2+1}=\frac{b}{b^2+1}\)

⇒ ab² + a = a²b + b

⇒ a - b = a2b - ab2

⇒ a - b = ab(a - b)

⇒ a = b OR ab = 1.

Since a = b is NOT the ONLY possibility, the given function is NEITHER One to One NOR Onto.