Relations & Functions
Bijective Functions
Grade 12
Question:
<p>Let \( f(x_1) = f(x_2) \), \( x_1, x_2 \in \mathbb{N} \) and \( f(x) = 4x + 3 \). Which of the following is true?</p>
<p>The function is one-one but not onto</p>
<p>The function is onto but not one-one</p>
<p>The function is neither one-one nor onto</p>
<p>The function is invertible (both one-one and onto)</p>
Step-by-Step Solution
Key Concept: A function is injective (one-to-one) if f(x₁) = f(x₂) implies x₁ = x₂. For linear functions with non-zero slope, this property always holds, so the given condition forces x₁ = x₂.
<p><strong>Step 1:</strong> Given f(x) = 4x + 3, where x₁, x₂ ∈ ℕ and f(x₁) = f(x₂).</p><p><strong>Step 2:</strong> From f(x₁) = f(x₂), we have: 4x₁ + 3 = 4x₂ + 3</p><p><strong>Step 3:</strong> Simplifying: 4x₁ = 4x₂ ⟹ x₁ = x₂</p><p><strong>Step 4:</strong> This proves that f is an injective (one-to-one) function. Therefore, the condition f(x₁) = f(x₂) with x₁, x₂ ∈ ℕ necessarily implies x₁ = x₂.</p><p><strong>Step 5:</strong> The true statement is: <strong>x₁ = x₂</strong> (or 'f is injective on ℕ')</p><p>∴ Answer: D</p>
Correct Answer: D