Solve rational equations, check for extraneous solutions, and identify asymptotes of simple rational functions.
To solve a rational equation, clear every denominator by multiplying both sides by the least common denominator -- this turns the equation into an ordinary polynomial equation you already know how to solve. The catch: multiplying by an expression containing the variable can introduce an EXTRANEOUS solution, a value that solves the cleared-out equation but makes an original denominator zero. Every solution must be checked against the original equation's domain.
The graph of a simple rational function like f(x) = 1/(x-h) + k has a vertical asymptote at x=h (where the function is undefined and shoots off toward infinity) and a horizontal asymptote at y=k (the value the function sneaks up on, but never reaches, as x grows huge in either direction).
Mastery looks like: They solve rational equations by clearing denominators, correctly reject any extraneous solution, and identify the asymptotes of a simple rational function.
Common stumbles: Forgetting to check a solution against the original domain; confusing which number gives the vertical asymptote versus the horizontal one.