Question

If \( f(x) = \frac{x}{1 - x} \), \( x \neq 1 \), then the inverse of \( f \) is:

• A. \( \frac{1 - x}{1 + x}, \, x \neq -1 \)
• B. \( \frac{1}{1 + x}, \, x \neq -1 \)
• C. \( \frac{1 - x}{x}, \, x \neq 0 \)
• D. \( \frac{x}{1 + x}, \, x \neq -1 \)
• E. \( \frac{1 + x}{1 - x}, \, x \neq 1 \)

Hint:

To find the inverse of a function, solve for \( x \) in terms of \( y \), and then replace \( y \) with \( x \).

Answer 1

The Correct Option is D

To find the inverse of \( f(x) = \frac{x}{1 - x} \), we solve for \( x \) in terms of \( y \): \[ y = \frac{x}{1 - x}. \] Multiplying both sides by \( 1 - x \) and solving for \( x \), we get: \[ y(1 - x) = x \quad \Rightarrow \quad y - yx = x \quad \Rightarrow \quad y = x(1 + y) \quad \Rightarrow \quad x = \frac{y}{1 + y}. \] Thus, the inverse function is \( f^{-1}(y) = \frac{y}{1 + y} \), where \( y \neq -1 \).