A function is defined by \(h : x \to 2 - \frac{1}{2x - 3}, x \neq \frac{3}{2}\). Find \(h^-1\), the inverse of h.

A function is defined by \(h : x \to 2 - \frac{1}{2x - 3}, x \neq \frac{3}{2}\). Find \(h^-1\), the inverse of h.

Answer Details

To find the inverse of a function, you need to switch the roles of x and y, then solve for y. The resulting expression will be the inverse function, denoted as \(h^{-1}\). Starting with the original function: \(h : x \to 2 - \frac{1}{2x - 3}, x \neq \frac{3}{2}\) Switching the roles of x and y: \(x = 2 - \frac{1}{2y - 3}\) Solving for y: \(x - 2 = - \frac{1}{2y - 3}\) \(-2y + 3 = -\frac{1}{x - 2}\) \(-2y = -\frac{1}{x - 2} - 3\) \(y = \frac{1}{2} \cdot \frac{1}{2 - x} + \frac{3}{2}\) \(y = \frac{3x - 7}{2x - 4}, x \neq 2\) Therefore, the inverse function is \(h^{-1} : x \to \frac{3x - 7}{2x - 4}, x \neq 2\). The correct option is (B).