Uploaded June 2022 | Updated September 2026, 2 weeks ago
A rational function is any function that can be written as a fraction whose numerator and denominator are polynomials. Rational functions include a broad range of possibilities. For example, since a polynomial can be a constant, a rational function's numerator and denominator can both be constants, in which case the rational function is simply a constant function. Unlike polynomial functions whose domains include all real numbers, in this lecture we show that any x value that causes the function's denominator to be zero will be excluded from the function's domain. These excluded x values produce missing points in the function's graph, and these missing points may be the locations of vertical asymptotes. Even though a function's graph can approach infinitely close to a vertical asymptote, it can never intersect that asymptote, since when the denominator's value is zero, the function's value is undefined. A vertical asymptote is therefore a vertical line that a function's graph can approach arbitrarily close to, but can never intersect.
A rational function is any function that can be written as a fraction whose numerator and denominator are polynomials. Rational functions include a broad range of possibilities. For example, since a polynomial can be a constant, a rational function's numerator and denominator can both be constants, in which case the rational function is simply a constant function. Unlike polynomial functions whose domains include all real numbers, in this lecture we show that any x value that causes the function's denominator to be zero will be excluded from the function's domain. These excluded x values produce missing points in the function's graph, and these missing points may be the locations of vertical asymptotes. Even though a function's graph can approach infinitely close to a vertical asymptote, it can never intersect that asymptote, since when the denominator's value is zero, the function's value is undefined. A vertical asymptote is therefore a vertical line that a function's graph can approach arbitrarily close to, but can never intersect.
