## rational function

### A rational function, also known as a rational expression, is a mathematical function in the form of a quotient of two polynomials

A rational function, also known as a rational expression, is a mathematical function in the form of a quotient of two polynomials. It can be represented as f(x) = p(x)/q(x), where p(x) and q(x) are polynomials and q(x) is not equal to zero.

The domain of a rational function is all real numbers except for the values of x that make the denominator equal to zero. These values are called the vertical asymptotes of the function since the function approaches infinity or negative infinity as x approaches these points.

To simplify a rational function, we can factor both the numerator and denominator polynomials and cancel out common factors. This is done to remove any removable discontinuities or holes in the graph. However, it is important to remember that cancelling out common factors should not be done if it results in making the denominator equal to zero.

Rational functions may have horizontal asymptotes, which are lines that the function approaches as x goes to positive or negative infinity. The behavior of the function near these asymptotes can be determined by comparing the degrees of the numerator and denominator polynomials.

Rational functions are important in various areas of mathematics and real-life applications, including physics, engineering, and economics. They provide a way to model and analyze relationships between different variables.

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