A rational function is the ratio of two polynomials. In other words, it’s a fraction where both the numerator and denominator are polynomial expressions.
The general form of a rational function is the following:
where P(x) and Q(x) are polynomials and Q(x) ≠ 0 (because division by zero is undefined).
Rational functions can have many features, including asymptotes, intercepts, and holes (points where the function is undefined due to a canceled factor). These functions are often more complex than linear or quadratic functions because the denominator can introduce vertical asymptotes and other discontinuities.
The domain of a rational function includes all real numbers except those that make the denominator equal to zero. Graphs of rational functions often display vertical asymptotes at those excluded values, and horizontal or slant (oblique) asymptotes that describe long-term behavior as x approaches infinity or negative infinity.
Consider the following rational function:
It simplifies to f(x) = x + 1 for all x except x = 1, where the function is undefined. This creates a hole in the graph at that point. Other rational functions, like f(x) = 1 / x, have vertical and horizontal asymptotes instead.
Rational functions are commonly studied in algebra and precalculus because they combine polynomial behavior with the complexity of division, producing rich and varied graphs.
For example, here is the graph of the rational function:
This function has both a vertical asymptote at x = -1 and a slant (or oblique) asymptote at y = x. The vertical asymptote occurs where the denominator is zero, and the function becomes undefined. The slant asymptote arises because the degree of the numerator is one higher than the degree of the denominator. As x becomes very large or very small, the graph closely follows the line y = x, showing the function’s long-term behavior.
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