Find the inverse function f⁻¹(x) for the function f(x) = (2x−1)/(x+2).

The given function is $f(x) = \frac{2x-1}{x+2}$.
Let $y = f(x)$, so we have:

$$y = \frac{2x-1}{x+2}$$

Multiply both sides by $(y+2)$:

$$x(y+2) = 2y-1$$

Distribute $x$ on the left side:

$$xy + 2x = 2y - 1$$

Gather all terms containing $y$ on one side of the equation and all other terms on the other side.
Subtract $2y$ from both sides and subtract $2x$ from both sides:

$$xy - 2y = -1 - 2x$$

Factor out $y$ from the terms on the left side:

$$y(x - 2) = -1 - 2x$$

Divide by $(x-2)$ to isolate $y$:

$$y = \frac{-1 - 2x}{x - 2}$$

This can also be written by multiplying the numerator and denominator by $-1$:

$$y = \frac{-(1 + 2x)}{-(2 - x)} = \frac{2x+1}{2-x}$$

$$f^{-1}(x) = \frac{2x+1}{2-x}$$

The inverse function is defined for all $x$ such that $2-x \neq 0$, which means $x \neq 2$.