Question:hard

The functions \(f(x)\) and \(g(x)\) are related as \(f(g(x)) = xg(f(f(x)))\), where \(f(x) = \dfrac{x}{x-1}\). What could be the functional form \(g(x)\)?

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First simplify \(f(f(x))\) using the given \(f(x)\); it collapses to a very simple expression.
Updated On: Jul 21, 2026
  • \(\dfrac{1}{x}\)
  • \(\dfrac{x}{x+1}\)
  • \(\dfrac{x+1}{x}\)
  • \(\dfrac{x}{x-1}\)
Show Solution

The Correct Option is C

Solution and Explanation

Step 1: Confirm that \(f(f(x))=x\).
Using \(f(x)=\dfrac{x}{x-1}\), applying \(f\) twice always returns \(x\), so the right side of the relation simplifies to \(xg(x)\).
This means testing candidates only needs the check \(f(g(x)) = xg(x)\).

Step 2: Pick a test value and evaluate each option.
Take \(x=3\). Option (a) gives \(g(3)=\dfrac{1}{3}\), option (b) gives \(g(3)=\dfrac{3}{4}\), option (c) gives \(g(3)=\dfrac{4}{3}\), and option (d) gives \(g(3)=\dfrac{3}{2}\).

Step 3: Check option (a).
\(f\left(\dfrac{1}{3}\right) = \dfrac{1/3}{1/3-1} = -\dfrac{1}{2}\), while \(xg(x) = 3 \times \dfrac{1}{3} = 1\). These do not match, so (a) fails.

Step 4: Check option (b) and option (d).
\(f\left(\dfrac{3}{4}\right) = -3\) against \(xg(x)=3\times\dfrac{3}{4}=2.25\), a mismatch, so (b) fails.
\(f\left(\dfrac{3}{2}\right) = 3\) against \(xg(x)=3\times\dfrac{3}{2}=4.5\), also a mismatch, so (d) fails.

Step 5: Check option (c).
\(f\left(\dfrac{4}{3}\right) = \dfrac{4/3}{4/3-1} = \dfrac{4/3}{1/3} = 4\), and \(xg(x) = 3 \times \dfrac{4}{3} = 4\). Both sides equal 4, so (c) holds.

Final Answer:
Only \(g(x)=\dfrac{x+1}{x}\) satisfies the relation for a test value, confirming option (c). \[ \boxed{g(x)=\dfrac{x+1}{x}} \]
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