Single correct · +4 / −1
A function f: ℝ → ℝ is defined by f(x) = x². It is:
- (a) One-one and onto
- (b) One-one but not onto
- (c) Onto but not one-one
- (d) Neither one-one nor onto
Answer and solution
Answer (d) Neither one-one nor onto
Solution
Maths sir · 3 steps
Test one-one with a pair like 2 and -2, and onto with a negative output like -1.
- One-one check. A one-one function never sends two different inputs to the same output. Here f(2) = 4 and f(-2) = 4, so two inputs share one output. So f is not one-one.
- Onto check. Onto means every number in the codomain ℝ is an output. A square is never negative, so x² = -1 has no real solution. So -1 is never reached.
Range of f = [0, ∞) ≠ ℝ
- So. The range is smaller than the codomain, so f is not onto either. The function is neither one-one nor onto, which is option (d).
The idea to keep. For one-one, look for two inputs with the same output. For onto, compare the range with the codomain. Always read the domain and codomain first: they decide the answer.
Answer from the NTA final answer key. Solution by PrepWiser.