Relations & Functions
Function evaluation
Grade 12
Question:
<p>If <em>x</em> = 2 + √3, and <em>f</em>(<em>x</em>) = (<em>x</em>² − 4<em>x</em> + 1)² + 2<em>x</em> − 2√3, find <em>f</em>(2 + √3).</p>
Step-by-Step Solution
Key Concept: Recognize that x = 2 + √3 satisfies a simple algebraic relation. Calculate x² - 4x + 1 first to simplify the nested expression before substituting into f(x).
<p><strong>Step 1:</strong> Given x = 2 + √3, first compute x² - 4x + 1.</p><p>x² = (2 + √3)² = 4 + 4√3 + 3 = 7 + 4√3</p><p>4x = 4(2 + √3) = 8 + 4√3</p><p>x² - 4x + 1 = (7 + 4√3) - (8 + 4√3) + 1 = 0</p><p><strong>Step 2:</strong> Since x² - 4x + 1 = 0, substitute into f(x):</p><p>f(2 + √3) = (0)² + 2(2 + √3) - 2√3</p><p>f(2 + √3) = 0 + 4 + 2√3 - 2√3</p><p>f(2 + √3) = 4</p><p>∴ Answer: <strong>4</strong></p>
Correct Answer: 4