JEE Main 2024 · 27 January · morning · Single correct · +4 / −1
The function f: N-{1} → N; defined by f(n)= the highest prime factor of n, is :
- (a) one-one only
- (b) neither one-one nor onto
- (c) onto only
- (d) both one-one and onto
Answer and solution
Answer (b) neither one-one nor onto
Solution
Maths sir · 4 steps
Find two numbers with the same highest prime factor, then find one output no input can reach.
- Test one-one. Take n = 2 and n = 4. The highest prime factor of both is 2, so f(2) = f(4) = 2. Two different inputs give the same output, so f is not one-one.
- Look at the outputs. For any n ≥ 2, f(n) is a prime number. So the range of f is the set of primes {2, 3, 5, 7, …}.
- Test onto. The codomain is all of N. Numbers like 1, 4 or 6 are in N but are not prime, so no n gives them as its highest prime factor. Range is not equal to codomain, so f is not onto.
- So. f is neither one-one nor onto, which is option (b).
The idea to keep. To kill one-one, find just one pair of inputs with the same output. To kill onto, find just one codomain element that can never be an output.
Answer from the NTA final answer key. Solution by PrepWiser.