Find the inverse of f(x)=(x−10)/(x+10).

검증된 단계별 안내

Start with the function equation: $y = \frac{x - 10}{x + 10}$.

To find the inverse, swap $x$ and $y$: $x = \frac{y - 10}{y + 10}$.

Multiply both sides by $(y + 10)$ to eliminate the denominator: $x(y + 10) = y - 10$.

Distribute $x$: $xy + 10x = y - 10$.

Group all terms involving $y$ on one side and factor $y$ out: $xy - y = -10 - 10x$, then $y(x - 1) = -10(1 + x)$, and finally solve for $y$: $y = \frac{-10(1 + x)}{x - 1}$.

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주요 개념

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Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. If f(x) maps x to y, then its inverse f⁻¹(x) maps y back to x. Finding an inverse involves solving the equation y = f(x) for x in terms of y.

Solving Rational Equations

Rational equations involve ratios of polynomials. To find the inverse of a rational function, you often need to solve for the variable by clearing denominators and isolating terms. This requires careful algebraic manipulation to avoid extraneous solutions.

Domain and Range Considerations

When finding inverses, it's important to consider the domain and range of the original function, as the inverse's domain and range are swapped. For rational functions, restrictions like division by zero must be accounted for to ensure the inverse is valid.