Inverse of a rational function

Definition

The inverse of a rational function is a function that reverses the effect of the original rational function. If $f(x)$ is a rational function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

5 Must Know Facts For Your Next Test

1. To find the inverse of a rational function, swap $x$ and $y$, then solve for $y$.
2. The domain of the original function becomes the range of its inverse and vice versa.
3. Not all rational functions have inverses; they must be one-to-one functions.
4. Horizontal asymptotes of the original function become vertical asymptotes in the inverse function.
5. Graphically, the function and its inverse are reflections across the line $y = x$.

Review Questions

• How do you determine if a rational function has an inverse?
• What steps are involved in finding the inverse of a given rational function?
• Explain how to check if two functions are inverses of each other.

Related terms

Rational Function: A ratio of two polynomials, typically expressed as $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

One-to-One Function: A function where each element of the range is paired with exactly one element of the domain. This property is crucial for a function to have an inverse.

Asymptote: A line that a graph approaches but never touches. Rational functions can have vertical, horizontal, or oblique asymptotes.