Sets, Relations & Functions
Functions
nta_pyq_2025_jan
Grade 11
Question:
The function f : (-\infty, \infty) \to (-\infty, 1), defined by f (x) = 2 -2 x -x is: 2 +2
Neither one-one nor onto
Onto but not one-one
Both one-one and onto
One-one but not onto
Step-by-Step Solution
Key Concept: Apply the core result for domains, ranges and functional equations and simplify using the given constraints.
2x 2 - 1 f (x) = (4) 2x 2 + 1 2 = 1 - 2x 2 + 1 2 ′ 2x f (x) = ⋅ 2.2 ⋅ ln 2 i.e always + ve 2 2x (2 + 1) so f (x) is ↑ function \therefore f (-\infty) = -1 f (\infty) = 1 \therefore f (x) \in (-1, 1) \ne co-domain so function is one-one but not onto
Correct Answer: 4