Question

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

• A. \(\frac{1}{x}\)
• B. \(\frac{x}{x+1}\)
• C. \(\frac{x+1}{x}\)
• D. \(\frac{x}{x-1}\)
• E. \(\frac{x}{1-x}\)

Answer 1

The Correct Option is C

1. Given the functional relationship: \( f(g(x)) = xg(f(f(x))) \), where \( f(x) = \frac{x}{x-1} \).
2. We need to find the form of \( g(x) \) that satisfies this equation.
3. First, substitute \( f(x) = \frac{x}{x-1} \) into the equation:
   • Calculate \( f(f(x)) \):
   • \( f(f(x)) = f\left(\frac{x}{x-1}\right) = \frac{\frac{x}{x-1}}{\frac{x}{x-1} - 1} = \frac{\frac{x}{x-1}}{\frac{x}{x-1} - \frac{x-1}{x-1}} \)
   • Continue simplifying:
     • \( = \frac{\frac{x}{x-1}}{\frac{x - (x-1)}{x-1}} = \frac{\frac{x}{x-1}}{\frac{1}{x-1}} = x \)
4. Now, substitute back into the given relationship:
   • \( f(g(x)) = x \cdot g(x) \)
   • This simplifies to: \( f(g(x)) = x \cdot g(x) \)
5. Find \( f(g(x)) \):
   • Assume \( g(x) = \frac{x+1}{x} \), then calculate \( f(g(x)) \):
   • \( f(g(x)) = f\left(\frac{x+1}{x}\right) = \frac{\frac{x+1}{x}}{\frac{x+1}{x} - 1} \)
   • Simplify the expression:
     • \( = \frac{\frac{x+1}{x}}{\frac{x+1-x}{x}} = \frac{\frac{x+1}{x}}{\frac{1}{x}} = x+1 \)
6. Verify if \( x \cdot g(x) \) equals \( x+1 \) when \( g(x) = \frac{x+1}{x} \):
   • \( x \cdot g(x) = x \cdot \frac{x+1}{x} = x+1 \)
   • This shows that the assumption \( g(x) = \frac{x+1}{x} \) is correct.
7. Thus, the correct functional form for \( g(x) \) is \(\frac{x+1}{x}\).

Therefore, the correct answer is indeed \( \frac{x+1}{x} \) .