Define a rational function.

A rational function is a function that can be expressed as the ratio of two polynomial functions.

A rational function is a function of the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions 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, which are called the vertical asymptotes. The range of a rational function depends on the degree of the numerator and denominator polynomials and the leading coefficients of those polynomials.

To simplify a rational function, we can factor the numerator and denominator and cancel out any common factors. We can also use long division to divide the numerator by the denominator and express the function as a polynomial plus a proper fraction. This can help us identify any horizontal asymptotes and end behavior of the function.

Rational functions can have various types of asymptotes, including vertical, horizontal, and slant asymptotes. Vertical asymptotes occur when the denominator of the function equals zero, and the function approaches positive or negative infinity as x approaches the value that makes the denominator zero. Horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator, and the function approaches a constant value as x approaches positive or negative infinity. Slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator, and the function approaches a linear function as x approaches positive or negative infinity.

Rational functions are used in many applications, including physics, engineering, and economics. They can model situations such as population growth, chemical reactions, and financial investments.