<p>Let <i>f</i> : (−∞, ∞) → [2, ∞) be a function defined by <i>f</i>(<i>x</i>) = <i>x</i>² − 2<i>a</i> + <i>a</i>², <i>a</i> ∈ ℝ. Find the value of <i>a</i> for which <i>f</i> is onto.</p>
Step-by-Step Solution
Key Concept: For a function to be onto, its range must equal the codomain. Set the minimum value of the quadratic equal to the lower bound of the codomain.
<p><strong>Step 1:</strong> For <i>f</i> to be onto, the range of the function should equal the codomain [2, ∞).</p><p><strong>Step 2:</strong> The minimum value of <i>f</i>(<i>x</i>) = <i>x</i>² − 2<i>a</i> + <i>a</i>² occurs at the vertex. Since the coefficient of <i>x</i>² is positive, the minimum value is <i>a</i>² − 2<i>a</i>.</p><p><strong>Step 3:</strong> For the range to be [2, ∞), the minimum value must equal 2.</p><p><i>a</i>² − 2<i>a</i> = 2</p><p><i>a</i>² − 2<i>a</i> − 2 = 0</p><p><strong>Step 4:</strong> Using the quadratic formula: <i>a</i> = $\frac{2 ± \sqrt{4 + 8}}{2}$ = $\frac{2 ± \sqrt{12}}{2}$ = $\frac{2 ± 2\sqrt{3}}{2}$ = 1 ± √5</p><p>∴ Answer is (d) 1 ± √5.</p>
Correct Answer: D