Relations & Functions
Even and Odd Functions
Grade 12
Question:
<p>If <span>\(f(x) = x^{11} + x^9 - x^7 + x^5 + x^3 + 1\)</span> and <span>\(f(\arcsin(\sin 8)) = a\)</span>, where <span>\(a\)</span> is a constant, then <span>\(f(\arctan(\tan 8))\)</span> is equal to</p>
<p>(a) <span>\(a\)</span></p>
<p>(b) <span>\(a - 2\)</span></p>
<p>(c) <span>\(a + 2\)</span></p>
<p>(d) <span>\(2 - a\)</span></p>
Step-by-Step Solution
Key Concept: The key property f(x) + f(-x) = 2 for this odd-like function enables us to relate values at different arguments.
<p><strong>Step 1:</strong> Note that <span>$f(x) + f(-x) = 2$</span>.</p><p><strong>Step 2:</strong> We have <span>$\arcsin(\sin 8) = 3\pi - 8 = y$</span> (say).</p><p><strong>Step 3:</strong> Since <span>$\arctan(\tan 8)$</span> has a specific relationship with the given angles and using the property <span>$f(x) + f(-x) = 2$</span>:</p><p><strong>Step 4:</strong> If <span>$f(\arcsin(\sin 8)) = a$</span>, then by the complementary property of the function, <span>$f(\arctan(\tan 8)) = 2 - a$</span>.</p><p>∴ Answer is (D).</p>
Correct Answer: D