Rational Functions and Simplification

Front

What is a rational function?

Back

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

Front

What is the domain of a rational function?

Back

The domain of a rational function is all real numbers except where the denominator equals zero.

Front

How do you find vertical asymptotes of a rational function?

Back

Vertical asymptotes occur at the values of x that make the denominator zero, provided the numerator is not also zero at those points.

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity.

Front

How do you determine the horizontal asymptote of a rational function?

Back

1. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. 2. If the degrees are equal, the horizontal asymptote is y = leading coefficient of numerator / leading coefficient of denominator. 3. If the degree of the numerator is greater, there is no horizontal asymptote.

Front

What is the y-intercept of a function?

Back

The y-intercept is the point where the graph of the function crosses the y-axis, found by evaluating the function at x = 0.

Front

How do you simplify a rational expression?

Back

To simplify a rational expression, factor both the numerator and the denominator and then cancel any common factors.

Front

What does it mean for a rational expression to be undefined?

Back

A rational expression is undefined when its denominator equals zero.

Front

How do you solve a rational equation?

Back

To solve a rational equation, find a common denominator, eliminate the denominators by multiplying through, and then solve the resulting equation.

Front

What is the significance of the speed of a plane in still air in wind problems?

Back

The speed of a plane in still air is the speed of the plane without any wind influence, used to set up equations for wind-related problems.