Sets, Relations & Functions
General
Grade None
Question:
The function \( f : \mathbb{R} \to \left[-\frac{1}{2}, \frac{1}{2}\right] \) defined as \( f(x) = \frac{x}{1+x^2} \) is :
neither injective nor surjective.
invertible.
injective but not surjective.
surjective but not injective.
Step-by-Step Solution
<div class="solution"><p>f(n)=2ⁿ. Sum=2ᵃ·Σ2ᵏ(k=1..10)=2ᵃ·2(2¹⁰-1)=16(2¹⁰-1) → 2ᵃ⁺¹=16=2⁴ → a=3.</p><p><strong>Answer: a=3</strong></p><div class="trap-box"><strong>Trap:</strong> f(x+y)=f(x)f(y) is multiplicative-on-addition, not f(xy).</div><div class="key-concept"><strong>Key Concept:</strong> Additive-input multiplicative f → f(n)=f(1)ⁿ</div></div>
Correct Answer: 4