Rational Function

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A rational function is a function that can be written as the quotient of two polynomials. The general form of a rational function is: f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials, and Q(x) ≠ 0. These functions have important characteristics such as zeros, asymptotes, and special points that determine their behavior on the number line.

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Graph of a Rational Function

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Rational Function 1

Asymptotes of a Rational Function

PROPERTIES OF A RATIONAL FUNCTION

DOMAIN

A rational function is defined for all real numbers except for those values of x for which the denominator Q(x) is equal to 0. For example, if Q(x) = x – 3, then the function is not defined for x = 3.

ZEROS OF THE FUNCTION

The zeros of the function are obtained by solving the equation P(x) = 0 (assuming Q(x) is not also zero at these points after simplification). This means that the zeros are the values of x for which the numerator of the function becomes zero.

VERTICAL ASYMPTOTES

Vertical asymptotes occur where the denominator Q(x) = 0 after any common factors with the numerator P(x) have been canceled out. At these x-values, the function is undefined and its graph tends towards infinity or negative infinity. For example, if f(x) = 1 / (x – 2), then the function has a vertical asymptote at x = 2.

HORIZONTAL ASYMPTOTES

Horizontal asymptotes depend on the degrees of the polynomials in the numerator and the denominator:

• If the degree of the numerator P(x) is less than the degree of the denominator Q(x), the horizontal asymptote is at y = 0.
• If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients of the polynomials.
• If the degree of the numerator is greater than the degree of the denominator, the function does not have a horizontal asymptote. (If the degree of P(x) is exactly one more than Q(x), there is a slant or oblique asymptote.)

EXAMPLES OF RATIONAL FUNCTIONS

SIMPLE RATIONAL FUNCTION:

f(x) = 1 / x

• Domain: x ≠ 0
• Vertical Asymptote: x = 0
• Horizontal Asymptote: y = 0

MORE COMPLEX FUNCTION (EXAMPLE REQUIRING SIMPLIFICATION):

f(x) = (x² – 4) / (x – 2) This function can be simplified: f(x) = [(x – 2)(x + 2)] / (x – 2) = x + 2, for x ≠ 2.

• Domain: x ≠ 2 (because the original denominator cannot be zero).
• Discontinuity: There is a "hole" or removable discontinuity at x = 2, not a vertical asymptote, because the (x-2) factor cancels out. The y-coordinate of the hole is 2+2=4.
• Horizontal Asymptote: Since the simplified function is linear (degree of numerator effectively becomes 1, degree of effective denominator is 0 after cancellation, or consider the original where degree of numerator (2) is greater than degree of denominator (1)), there is no horizontal asymptote.

CONCLUSION

Rational functions have wide application in mathematics as they allow for the analysis of complex relationships between variables. Their zeros, asymptotes, and domains are key elements in studying their graphs and behavior in various mathematical contexts. Understanding these functions is fundamental for further studies in analysis and algebra.