Sketch a rational function using its intercepts and asymptotes, and solve rational equations and inequalities algebraically.
Graphing a rational function completely means finding four things first: the domain (and any holes), every vertical asymptote, the horizontal or slant asymptote, and the intercepts (set the numerator to 0 for x-intercepts, evaluate at x=0 for the y-intercept). Once those pieces are in place, the curve has to connect them the only way that respects every asymptote as a boundary it approaches but never crosses in that vicinity.
Solving a rational EQUATION means clearing every denominator (multiply both sides by it) and solving the resulting polynomial equation -- then checking that no solution accidentally makes an original denominator 0. Solving a rational INEQUALITY is different: you cannot simply multiply both sides by an expression whose sign you do not know. Instead, move everything to one side, find every critical value (zeros of the numerator AND the denominator), and test the sign of the whole expression in each interval those values create.
Mastery looks like: They find all the graphing pieces of a rational function, solve a rational equation while checking for extraneous solutions, and solve a rational inequality using a sign chart.
Common stumbles: Forgetting to check a rational equation's solution against the original denominators; multiplying a rational inequality by an expression as if it were always positive.