Sets, Relations & Functions
Highest Prime Factor Function — Injectivity/Surjectivity
nta_pyq_2024_jan
Grade 11
Question:
The function $f:\mathbb{N}-\{1\}\to\mathbb{N}$; defined by $f(n)=$ the highest prime factor of $n$, is:
both one-one and onto
one-one only
onto only
neither one-one nor onto
Step-by-Step Solution
Key Concept: Check one-one: $f(2)=f(4)=2$ (same output for different inputs). Check onto: $4\in\mathbb{N}$ but $4$ is not the highest prime of any natural number (primes are always prime, not composite). So not onto.
$f(2)=f(4)=2$: many-one. $4\in\mathbb{N}$ is not a prime and not the highest prime of any number: not onto. Neither one-one nor onto.
Correct Answer: 4