Exploring the Key Features of Rational Functions | Asymptotes, Intercepts, and More

rational function

A rational function is a function that can be expressed as the quotient of two polynomial functions, where the denominator polynomial is not zero

A rational function is a function that can be expressed as the quotient of two polynomial functions, where the denominator polynomial is not zero. In other words, it is a function in the form:

f(x) = \frac{P(x)}{Q(x)}

where P(x) and Q(x) are polynomial functions, and Q(x) is not zero for any value of x.

The domain of a rational function includes all values of x except for those that make the denominator zero. These values, called the vertical asymptotes, are found by setting the denominator equal to zero and solving for x.

The graph of a rational function can have several features, including horizontal and vertical asymptotes, x-intercepts, and holes. The vertical asymptotes occur at the values of x that make the denominator zero. The horizontal asymptotes can be found by comparing the degrees of the numerator and denominator polynomials. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is a horizontal line defined by the ratio of the leading coefficients of the numerator and denominator. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

To graph a rational function, it is helpful to analyze its behavior near the vertical and horizontal asymptotes. As x gets very large (positive or negative), the function approaches its horizontal asymptote. As x approaches the values that make the denominator zero, the function may have vertical asymptotes or holes. By plotting points, considering the behavior near the asymptotes, and analyzing the end behavior, the graph of a rational function can be sketched.

It is important to note that rational functions can have asymptotes, intercepts, and other key features, making them useful in various applications such as physics, engineering, and economics.

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