What Is a Rational Function?

A rational function is a function written as one polynomial divided by another: f(x) = p(x) ÷ q(x), where q(x) is not the zero polynomial. Because you can never divide by zero, any x-value that makes the denominator zero is excluded from the domain.

f(x) = 1/x is the most basic rational function. Its domain is every real number except x = 0 (since 1/0 is undefined). The graph has two separate branches — one where x is positive, one where x is negative — that never touch, approaching a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.

To find a rational function's domain, set the denominator equal to zero and solve — those x-values are excluded. For f(x) = 1/(x−3), the domain excludes x = 3. Some functions exclude more than one value, or the denominator may factor to reveal several restrictions at once.

Not every excluded value looks the same on the graph. Some create a removable discontinuity — a single-point hole — if the factor causing the zero cancels out algebraically. Others create a true vertical asymptote, where the graph shoots toward infinity instead. Once you can graph the function, you can also study its behavior with the full graphs of rational functions.

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