Relations & Functions
Functional Equations
Grade 12
Question:
<p>If <span style='font-style:italic;'>a</span> + <span style='font-style:italic;'>a</span> = 1, <span style='font-style:italic;'>b</span> + <span style='font-style:italic;'>b</span> = 2 and <span style='font-style:italic;'>af</span>(<span style='font-style:italic;'>x</span>) + <span style='font-style:italic;'>af</span>\(\left(\frac{1}{x}\right)\) = <span style='font-style:italic;'>bx</span> + \(\frac{b}{x}\), <span style='font-style:italic;'>x</span> ≠ 0, then the value of the expression \(\frac{f(x) + f\left(\frac{1}{x}\right)}{x + \frac{1}{x}}\) is</p>
Step-by-Step Solution
Key Concept: Use the given functional equation with two different substitutions (x and 1/x) to create a system of linear equations in f(x) and f(1/x), then solve for the required expression.
<p><strong>Step 1: Interpret the given conditions</strong></p><p>From 'a + a = 1', we get 2a = 1, so <strong>a = 1/2</strong></p><p>From 'b + b = 2', we get 2b = 2, so <strong>b = 1</strong></p><p><strong>Step 2: Write the functional equation</strong></p><p>The given equation is: af(x) + af(1/x) = bx + b/x</p><p>Substituting a = 1/2 and b = 1:</p><p><strong>(1/2)f(x) + (1/2)f(1/x) = x + 1/x</strong> ... (Equation 1)</p><p><strong>Step 3: Substitute x → 1/x in Equation 1</strong></p><p>Replace x with 1/x in the functional equation:</p><p>(1/2)f(1/x) + (1/2)f(x) = 1/x + x</p><p><strong>(1/2)f(1/x) + (1/2)f(x) = x + 1/x</strong> ... (Equation 2)</p><p><strong>Step 4: Observe that Equations 1 and 2 are identical</strong></p><p>Both equations give us: (1/2)f(x) + (1/2)f(1/x) = x + 1/x</p><p>Multiply both sides by 2:</p><p><strong>f(x) + f(1/x) = 2x + 2/x</strong> ... (Equation 3)</p><p><strong>Step 5: Calculate the required expression</strong></p><p>We need to find: [f(x) + f(1/x)] / [x + 1/x]</p><p>From Equation 3: f(x) + f(1/x) = 2x + 2/x = 2(x + 1/x)</p><p>Therefore:</p><p>[f(x) + f(1/x)] / [x + 1/x] = 2(x + 1/x) / (x + 1/x) = <strong>2</strong></p><p><strong>∴ Answer: 2</strong></p>
Correct Answer: 2