Inverse of a rational function

Definition

The inverse of a rational function is a function that reverses the effect of the original rational function. It essentially swaps the dependent and independent variables, solving for the input in terms of the output.

5 Must Know Facts For Your Next Test

1. The inverse exists only if the original function is one-to-one (bijective).
2. You find the inverse by swapping $x$ and $y$ in the equation and then solving for $y$.
3. The domain of the original function becomes the range of the inverse, and vice versa.
4. For a rational function $f(x) = \frac{P(x)}{Q(x)}$, its inverse is usually found by algebraic manipulation.
5. Graphically, the graph of an inverse function is a reflection over the line $y = x$.

Review Questions

Related terms

Rational Function:

A function represented by a ratio of two polynomials, i.e., $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials.

One-to-One Function:

A function where each input corresponds to exactly one output, ensuring it has an inverse.

Domain:

The set of all possible input values ($x$-values) for which a function is defined.