In Exercises 41–44:
a. Find f⁻¹(x).


42. f(x) = (x + 2) / (1 − x), a = 1/2

Verified step by step guidance

1. Start by writing the function as an equation with y: \(y = \frac{x + 2}{1 - x}\).
2. To find the inverse function \(f^{-1}(x)\), swap the roles of \(x\) and \(y\): \(x = \frac{y + 2}{1 - y}\).
3. Multiply both sides of the equation by the denominator \((1 - y)\) to eliminate the fraction: \(x(1 - y) = y + 2\).
4. Distribute \(x\) on the left side: \(x - xy = y + 2\).
5. Group all terms involving \(y\) on one side and constants on the other: \(x - 2 = y + xy\). Then factor \(y\) out: \(x - 2 = y(1 + x)\). Finally, solve for \(y\): \(y = \frac{x - 2}{1 + x}\), which is \(f^{-1}(x)\).

Key Concepts

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. To find f⁻¹(x), solve the equation y = f(x) for x in terms of y, then interchange x and y. The inverse exists only if the function is one-to-one (bijective) on the domain considered.

Solving Rational Equations

Finding the inverse of a rational function often requires solving equations involving fractions. This involves clearing denominators, isolating variables, and simplifying expressions carefully to avoid extraneous solutions or domain issues.

Domain and Range Considerations

When finding inverses, it is crucial to consider the domain of the original function and the range of the inverse. Restrictions on x (like denominators not being zero) affect the domain, and the inverse’s domain corresponds to the original function’s range.