<p>A function \(f(x)\) which is invertible must be</p>
Step-by-Step Solution
Key Concept: A function is invertible if and only if it is bijective (both injective and surjective). This means each element in the codomain must map to exactly one element in the domain, which requires the function to be both one-one and onto.
<p><strong>Step 1:</strong> Recall that a function f: A → B is invertible if and only if there exists a function g: B → A such that g(f(x)) = x for all x ∈ A and f(g(y)) = y for all y ∈ B.</p><p><strong>Step 2:</strong> This is possible if and only if f is bijective, meaning:</p><ul><li><strong>Injective (One-one):</strong> f(x₁) = f(x₂) ⟹ x₁ = x₂ (no two distinct elements map to the same element)</li><li><strong>Surjective (Onto):</strong> For every y in the codomain, there exists x in the domain such that f(x) = y (every element in codomain is mapped to)</li></ul><p><strong>Step 3:</strong> Without both properties, an inverse function cannot exist. If f is not one-one, the inverse would be multi-valued. If f is not onto, the inverse would be undefined for some elements in the codomain.</p><p>∴ A function which is invertible must be <strong>bijective (both one-one and onto)</strong></p>
Correct Answer: B