In Exercises 41–44:
a. Find f⁻¹(x).

42. f(x) = (x + 2) / (1 − x), a = 1/2

Guida verificata passo dopo passo

Start by writing the function as an equation with y: $y = \frac{x + 2}{1 - x}$.

To find the inverse function $f^{-1}(x)$, swap the roles of $x$ and $y$: $x = \frac{y + 2}{1 - y}$.

Multiply both sides of the equation by the denominator $(1 - y)$ to eliminate the fraction: $x(1 - y) = y + 2$.

Distribute $x$ on the left side: $x - xy = y + 2$.

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 $f^{-1}(x)$.

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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.

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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.

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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.

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Percorso guidato