Sets, Relations & Functions
General
Grade 11
Question:
Let \( \mathbb{N} \) be the set of natural numbers and two functions f and g be defined as f,g : \( \mathbb{N} \to \mathbb{N} \) such that : \( f(n) = \frac{n + 1}{2} \) if n is odd and g(n) = n - (-1)n. The fog is :
Both one-one and onto
One-one but not onto
Neither one-one nor onto
onto but not one-one
Step-by-Step Solution
<div class="solution"><p>Observe g(2x-3)=4x²-10x+5. So f(x)=2x-3 or 2-2x. At x=5/4: both give -1/2. Answer: -1/2.</p><div class="trap-box"><strong>Trap:</strong> Even if g not one-one, the asked value can still be unique at a specific point.</div><div class="key-concept"><strong>Key Concept:</strong> Reverse-engineer g rather than solving quadratic in f(x)</div></div>
Correct Answer: 4