Function domain and range
Find the exact domain of a function and, for polynomial and rational functions, its exact range. Excluded values and irrational endpoints such as sqrt(2) are shown exactly.
Functions are limited to 1,000 characters, one variable, 24 nesting levels, 100 digits per number, 4 abs terms and polynomial factors up to degree 24. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of points. Use one variable, ^ for powers and functions such as sqrt(x) or ln(x). Up to 1,000 characters. Review the settings before running.
Your result
How to use Function domain and range
- Type the function, such as (x^2 - 1)/(x^2 - 4).
- Check the variable name.
- Run the calculation and read the domain, the range and the listed endpoints.
- Download the JSON report or the CSV of points.
Useful next steps
Functions are limited to 1,000 characters, one variable, 24 nesting levels, 100 digits per number, 4 abs terms and polynomial factors up to degree 24. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of points.
Domain
The tool collects every condition the expression needs: denominators not equal to zero, square roots and fractional powers of nonnegative values, logarithms of positive values, and asin or acos of values between -1 and 1. It solves them exactly with Sturm sequences and sign charts. Method: OpenStax, College Algebra 2e, section 3.2 Domain and Range (2021); Basu, Pollack and Roy, Algorithms in Real Algebraic Geometry, chapter 2 (2006).
Range
For polynomial and rational functions, including abs terms, the real line is split at critical points, excluded values and domain endpoints. On each piece the function is monotone, so the range is built from exact values and limits. Values that would need roots of degree 3 or higher are refused instead of guessed.
What is refused
Ranges of functions with sqrt, logarithms, trigonometric functions, pi or e are not computed, and domains with infinitely many exclusions such as tan(x) are rejected. The report says why.