Rational (Reciprocal) Functions: Properties And Examples

Rational (Reciprocal) Function

y=1/x, has an asymptote at zero (because the function is undefined if we divide by 0).

A rational (reciprocal) function is a function of the form f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials, and Q(x) is not equal to zero. This type of function is called reciprocal because it involves the reciprocal of x, or 1/x, in its terms.

Some examples of rational (reciprocal) functions include:

– f(x) = 1/x
– f(x) = (x + 2) / (x – 3)
– f(x) = (x^2 + 1) / (3x – 5)

Properties of Rational (Reciprocal) Functions:

1. Domain: The domain of a rational function is all real numbers except the values of x that make the denominator Q(x) equal to zero. These values are called the vertical asymptotes of the function.

2. Range: The range of a rational function can be found by analyzing the behavior of the function near its vertical asymptotes. If the degree of the numerator polynomial P(x) is less than the degree of the denominator polynomial Q(x), then the range of the function is all real numbers. If the degree of P(x) is greater than or equal to the degree of Q(x), then the range of the function is either all negative real numbers or all positive real numbers, depending on the sign of the leading coefficient of P(x).

3. Horizontal asymptotes: If the degree of the numerator polynomial P(x) is less than the degree of the denominator polynomial Q(x), then the horizontal asymptote of the function is y = 0. If the degree of P(x) is equal to the degree of Q(x), then the horizontal asymptote of the function is y = the ratio of the leading coefficients of P(x) and Q(x). If the degree of P(x) is greater than the degree of Q(x), then the function has no horizontal asymptote.

4. Symmetry: Rational functions are generally not symmetric about the y-axis, but they may have symmetry about the origin or the x-axis, depending on the form of the function.

Overall, rational (reciprocal) functions are important in mathematical modeling because they can be used to describe various relationships between variables, such as rates of change, growth, and decay. Understanding the properties of these functions is fundamental to many fields of study, including economics, physics, and engineering.

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