About the Domain and Range Finder for Rational Functions

A rational function is defined as the ratio of two polynomials. For high school algebra, we often focus on the linear-over-linear form f(x) = (ax+b)/(cx+d). Analyzing these functions requires identifying values that “break” the math, specifically division by zero.

Finding the Domain

The Domain is the set of all possible x-values (inputs). For rational functions, the domain includes all real numbers except those that make the denominator zero.
To find it: Set cx + d = 0 and solve for x. The solution is the Vertical Asymptote, and it must be excluded from the domain.

Finding the Range

The Range is the set of all possible y-values (outputs). For this specific type of rational function, as x gets infinitely large (positive or negative), the function approaches the ratio of the leading coefficients: y = a/c.
This value represents the Horizontal Asymptote. The graph gets closer and closer to this y-value but usually never touches it. Thus, the range is all real numbers except y = a/c.

Why use this Domain and Range Finder for Rational Functions?

Visualizing asymptotes is difficult when drawing by hand. This tool calculates the exact values for the exclusions and graphs them as dashed lines, helping students connect the algebraic restrictions to the geometric behavior of the hyperbola.