Types of Rational Function

Any function that can be defined by a rational fraction, which is an algebraic fraction in which both the numerator and denominator are polynomials, is known as a rational function. Polynomial coefficients do not have to be rational numbers; they might be in any field K. The rational function and the rational fraction over K is used in this situation. The variables’ values can be found in any field L containing K. The function’s domain is the set of variables’ values for which the denominator is not zero, and the codomain is L. The field of fractions of the ring of polynomial functions is the set of rational functions over a field K.

Rational Function

A rational function is one that can be represented as the quotient or ratio of two polynomial functions, with the denominator polynomial having at least one degree. Rational functions are important for practical techniques like the visual depiction of a problem since they arise naturally in many circumstances. 

Formula for Rational Functions

A reasonable procedure R (f) is a function of type A(f)/B(f), where A(f) and B(f) are polynomial functions and B(f) is a non-zero polynomial.

R(f) =A(f)*B(f)*R(f) = A(f)*B(f), where B(f)≠0 is the rational function formula.

every polynomial function is a rational function, a rational function becomes a polynomial function only when B (f) = 1, i.e. when it is a constant polynomial function.

A function that cannot be represented in the form of a polynomial, such as R(f) = sin(f), is not a rational function, according to the rational function definition. Constant functions, such R(f) = pi, are rational value functions since they are polynomials. Even if the value of R(f) is irrational for all f-values, the function itself is rational.

Rational Functions Types

The types of Asymptotes found in the graphing of rational functions determine the types of rational functions. The axis of the independent variable is parallel to the horizontal asymptote (parallel to the x-axis). Horizontal asymptotes are horizontal lines that appear on the graph of a rational function when it approaches +infinity or –infinity. Vertical Asymptotes are vertical lines that are parallel to the y-axis and near which the function can expand indefinitely. There are no vertical or horizontal asymptotes in an oblique asymptote. As ‘f’ approaches +infinity or –infinity, the oblique or slant asymptotes are the diagonal lines in the curve that approaches zero.

• Rational Function Asymptotes: Asymptotes for Rational Functions: There are three types of asymptotes for a rational function: horizontal, vertical, and slant asymptotes. In addition to this, it might have holes. Let’s see how to find each of them.

• Holes in a Rational Function:

The holes in a rational function’s graph are points that appear to be present on the graph but are not. Solving for x and setting the linear factors that are common factors of both the numerator and denominator of the function to zero will reveal them. We may acquire the relevant y-coordinates of the spots by entering the x-values into the simplified function. Any rational function does not require any holes. Holes emerge only when the numerator and denominator have linear common factors.

• Vertical Asymptote of a Rational Function:

The vertical asymptote (VA) of a function is a hypothetical vertical line that the graph approaches but never crosses. The formula is x = a number. The domain’s disallowed values are directly related to “some number.” However, there can’t be a vertical asymptote at x = some integer if there’s a hole there. Follow these procedures to get the vertical asymptotes of a rational function: To eliminate all common components, first simplify the function (if any). Find (x) by setting the numerator to 0.

• Horizontal Asymptote of a Rational Function:

A horizontal asymptote is an imaginary horizontal line to which the graph of a function appears to be extremely close but never touches (HA). The formula is y = a number. The omitted values in the range are directly related with “some number.” For a rational function, there can only be one horizontal asymptote option. The horizontal asymptote of a rational function may be found using the degrees of the numerator (N) and denominators (D) (D).

If N D, then there is a HA at y = 0.

If N is more than D, there is no HA.

If N = D, the HA is y = ratio of the leading coefficients.

• Rational Function Slant (Oblique) Asymptotes: 

An imaginary oblique line that appears to touch a piece of the graph is called a slant asymptote. A rational function can only have a slant asymptote when the degree of the numerator (N) is precisely one greater than the degree of the denominator (D). Its equation is y = quotient, which is obtained by dividing the numerator by the denominator using long division.

Conclusion

A rational function is one in which the ratio of polynomials is the same. A rational function is one with only one variable, x, and may be written as f(x) = p(x)/q(x), where p(x) and q(x) are polynomials with q(x) ≠ 0.

Because every constant is a polynomial, the numerators of a rational function can also be constants. F(x) = 1/(3x+1) is an example of a rational function. The denominators of rational functions, however, cannot be constants. For instance, f(x) = (2x + 3) / 4 is a linear function, not a rational function.