Question:medium

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)?

Updated On: Jan 13, 2026
  • \(\frac{1}{x}\)
  • \(\frac{x}{x+1}\)
  • \(\frac{x+1}{x}\)
  • \(\frac{x}{x-1}\)
  • \(\frac{x}{1-x}\)
Show Solution

The Correct Option is C

Solution and Explanation

The correct answer is option (C):
\(\frac{x+1}{x}\)

Let's break down this problem step by step to find the correct functional form for g(x).

We are given two pieces of information:

1. f(g(x)) = xg(f(f(x)))
2. f(x) = x / (x - 1)

Our goal is to find g(x). We'll do this by substituting and simplifying.

First, let's find f(f(x)). We substitute f(x) into the function f(x):

f(f(x)) = f(x / (x - 1)) = (x / (x - 1)) / ((x / (x - 1)) - 1)

Now, let's simplify the expression:

f(f(x)) = (x / (x - 1)) / ((x - (x - 1)) / (x - 1))
f(f(x)) = (x / (x - 1)) / (1 / (x - 1))
f(f(x)) = x

Now, let's substitute f(f(x)) = x into the main equation:

f(g(x)) = xg(f(f(x)))
f(g(x)) = xg(x)

We also know that f(x) = x / (x - 1). Therefore, to find f(g(x)), we substitute g(x) into f(x):

f(g(x)) = g(x) / (g(x) - 1)

Now we can set up the equation:

g(x) / (g(x) - 1) = xg(x)

We need to solve for g(x). If g(x) is not equal to 0, we can divide both sides by g(x):

1 / (g(x) - 1) = x

Now, solve for g(x):

1 = x(g(x) - 1)
1 = xg(x) - x
x + 1 = xg(x)
g(x) = (x + 1) / x

Therefore, the correct answer is g(x) = (x + 1) / x. This is the only functional form that satisfies the given conditions.
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