<p>If \(f(g(x)) = \ln\left(\dfrac{(1+x)^3}{(1-x)^3}\right) = 3f(x)\), then which of the following is true?</p>
<p>\(f(x) = \ln\left(\dfrac{1-x}{1+x}\right)\)</p>
<p>\(f(x) = \ln\left(\dfrac{1+x}{1-x}\right)\)</p>
<p>\(f(x) = \ln\left(\dfrac{(1+x)^3}{(1-x)^3}\right)\)</p>
<p>\(f(x) = 3\ln\left(\dfrac{1+x}{1-x}\right)\)</p>
Step-by-Step Solution
Key Concept: Recognize that f(g(x)) = 3f(x) means g(x) must transform the input in a way that scales the logarithmic output by factor 3. Since ln(a³) = 3ln(a), we need g(x) to produce the argument of f that, when cubed, gives our expression.
<p><strong>Step 1:</strong> Simplify the given expression using logarithm properties.</p><p>f(g(x)) = ln⎜(1+x)³/(1-x)³⎜ = 3ln|(1+x)/(1-x)|</p><p><strong>Step 2:</strong> Since f(g(x)) = 3f(x), we have:</p><p>3ln|(1+x)/(1-x)| = 3f(x)</p><p>∴ f(x) = ln|(1+x)/(1-x)|</p><p><strong>Step 3:</strong> This means g(x) must equal x (identity function), making:</p><p>f(g(x)) = f(x) = ln|(1+x)/(1-x)| = 3f(x) only if we reconsider...</p><p><strong>Step 3 (Corrected):</strong> Actually, if f(g(x)) = 3f(x), then g(x) = x³, so:</p><p>f(x³) = 3f(x), which means f(x) = ln|x| and g(x) = (1+x)/(1-x)</p><p>Verification: f(g(x)) = ln|(1+x)/(1-x)| = 3ln|(1+x)^(1/3)/(1-x)^(1/3)| = 3f(x) ✓</p><p><strong>∴ Answer: B</strong></p>
Correct Answer: B