A function f is defined on R, the set of real numbers, by: \(f : x \to \frac{x + 3}{x - 2}, x \neq 2\), find \(f^{-1}\).

A function f is defined on R, the set of real numbers, by: \(f : x \to \frac{x + 3}{x - 2}, x \neq 2\), find \(f^{-1}\).

Answer Details

To find \(f^{-1}\), we need to solve the equation: \[y = \frac{x + 3}{x - 2}\] for x in terms of y. To do this, we start by swapping the roles of x and y: \[x = \frac{y + 3}{y - 2}\] Now we solve for y in terms of x. We begin by multiplying both sides by \(y-2\): \[x(y-2) = y + 3\] Expanding the left-hand side: \[xy - 2x = y + 3\] Rearranging the terms: \[xy - y = 2x + 3\] Factoring out y on the left-hand side: \[y(x-1) = 2x + 3\] Finally, dividing both sides by \(x-1\): \[y = \frac{2x+3}{x-1}, x\neq 1\] Therefore, the inverse function of \(f(x) = \frac{x+3}{x-2}, x\neq 2\) is: \[f^{-1}(x) = \frac{2x+3}{x-1}, x\neq 1\] Hence, the answer is \(\boxed{\textbf{(A) }f^{-1} : x \to \frac{2x+3}{x-1}, x \neq 1}\).