<p>If the function <i>f</i> : ℝ − {1, −1} → <i>A</i> defined by <i>f</i>(<i>x</i>) = \(\frac{x^2}{1-x^2}\) is surjective, then <i>A</i> is equal to</p>
Step-by-Step Solution
Key Concept: For a function to be surjective (onto), every element in the codomain must be in the range. Express <i>y</i> in terms of <i>x</i> and determine which values of <i>y</i> are achievable.
<p><strong>Step 1:</strong> Given function <i>f</i>(<i>x</i>) = $\frac{x^2}{1-x^2}$ = <i>y</i> (let).</p><p><strong>Step 2:</strong> Solve for <i>x</i>² in terms of <i>y</i>:</p><p><i>x</i>² = <i>y</i>(1 − <i>x</i>²)</p><p><i>x</i>² = <i>y</i> − <i>y</i><i>x</i>²</p><p><i>x</i>²(1 + <i>y</i>) = <i>y</i></p><p><i>x</i>² = $\frac{y}{1+y}$ (provided <i>y</i> ≠ −1)</p><p><strong>Step 3:</strong> Since <i>x</i>² ≥ 0, we require:</p><p>$\frac{y}{1+y}$ ≥ 0</p><p><strong>Step 4:</strong> Analyze the sign: This inequality holds when <i>y</i> ∈ (−∞, −1) ∪ [0, ∞).</p><p><strong>Step 5:</strong> For a surjective function, the range equals the codomain. Therefore, <i>A</i> = ℝ − [−1, 0).</p><p>∴ Answer is (c) ℝ − [−1, 0).</p>
Correct Answer: C