Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 20: Special Functions

Lesson 4: Rational Functions

In this Grade 4 AMC Math lesson from AoPS: Introduction to Algebra, students learn to define and work with rational functions — functions expressed as the ratio of two polynomials. The lesson covers solving rational equations by clearing denominators, identifying the domain and range of a rational function, and understanding horizontal and vertical asymptotes. Students practice these concepts through problems involving functions like f(x) = (3x − 4)/(x + 5), exploring how the function behaves as x approaches restricted values or grows very large.

Section 1

Rational Functions - Definition and Domain

Property

A rational function is a function of the form R(x)=p(x)q(x)R(x) = \frac{p(x)}{q(x)} where p(x)p(x) and q(x)q(x) are polynomial functions and q(x)≠0q(x) \neq 0. The domain of a rational function is all real numbers except for those values that make the denominator equal to zero, i.e., where q(x)=0q(x) = 0.

Examples

Section 2

Rational Functions and Their Asymptotes

Property

A rational function has the form f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)} where P(x)P(x) and Q(x)Q(x) are polynomials and Q(x)≠0Q(x) \neq 0. An asymptote is a line that the graph of a function approaches but never touches. Rational functions can have vertical asymptotes (where the denominator equals zero) and horizontal asymptotes (determined by the behavior as xx approaches infinity).

Examples

Section 3

Finding Range of Rational Functions

Property

To find the range of a rational function f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)}, set y=f(x)y = f(x) and solve for xx in terms of yy: x=g(y)x = g(y). The range consists of all yy-values for which this equation has valid solutions (where denominators are non-zero).

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Rational Functions - Definition and Domain

Property

A rational function is a function of the form R(x)=p(x)q(x)R(x) = \frac{p(x)}{q(x)} where p(x)p(x) and q(x)q(x) are polynomial functions and q(x)≠0q(x) \neq 0. The domain of a rational function is all real numbers except for those values that make the denominator equal to zero, i.e., where q(x)=0q(x) = 0.

Examples

Section 2

Rational Functions and Their Asymptotes

Property

A rational function has the form f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)} where P(x)P(x) and Q(x)Q(x) are polynomials and Q(x)≠0Q(x) \neq 0. An asymptote is a line that the graph of a function approaches but never touches. Rational functions can have vertical asymptotes (where the denominator equals zero) and horizontal asymptotes (determined by the behavior as xx approaches infinity).

Examples

Section 3

Finding Range of Rational Functions

Property

To find the range of a rational function f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)}, set y=f(x)y = f(x) and solve for xx in terms of yy: x=g(y)x = g(y). The range consists of all yy-values for which this equation has valid solutions (where denominators are non-zero).

Examples