Qué es una función racional

What is a Rational Function?

Definition and Basic Concept

• A rational function is defined as f(x) = p(x)/q(x) , where p(x) and q(x) are polynomials, and q(x) neq 0 .

• The key aspect of rational functions is that the denominator must not equal zero to avoid indeterminate forms in division.

Examples of Rational Functions

• An example of a basic rational function is f(x) = 1/x , which has a degree 0 polynomial in the numerator and a degree 1 polynomial in the denominator.

• The graph of this function approaches but never touches the horizontal asymptote at y = 0, indicating that x cannot be zero. This creates a vertical asymptote at x = 0.

Asymptotes in Rational Functions

• Vertical asymptotes occur when the denominator equals zero; for instance, if g(x) = 1/x + 5 , there’s a vertical asymptote at x = -5.

• Horizontal asymptotes can also exist; for example, if the horizontal line passes through y = 1, it indicates that as x approaches infinity or negative infinity, the function approaches this value.

Higher Degree Polynomials

• Another example includes functions like h(x) = x^2/x - 3 , which features both vertical (at x = 3) and potentially oblique asymptotes due to its higher degree numerator compared to its linear denominator.

• Oblique asymptotes arise when the degree of the numerator exceeds that of the denominator by one; they represent slanting lines rather than horizontal ones.

Polynomial Functions as Rational Functions

• It’s noted that all polynomial functions are also considered rational functions since dividing by a constant (degree 0 polynomial) yields another polynomial expression. However, these cases are typically excluded from discussions on rational functions unless specified otherwise.

• In practice, rational functions are primarily studied when their denominators have degrees of one or higher to explore their unique characteristics effectively.